When did the Hindu temple culture start in India?

First of all, which is the oldest Hindu temple that still exists?

 I haven't been able to find any one temple that Historians have labelled as the oldest temple. In my view, the Dashavatara Vishnu Temple (meaning "ten incarnations") located at Deogarh, Uttar Pradesh in Central India built around 500 CE may be one of the earliest Hindu stone temples still surviving today. Picture below -

As the OP has mentioned, the Buddhist Stupas at Sanchi or Bharhut or Dhamek, have origins that are 700 years earlier (Ashoka's reign around 250 BC) than the oldest temples.

Did the Hindu Temple culture exist prior to 500 AD? Is it a borrowed concept from Buddhist Stupas? The answer to these questions involves some historical evidence but significant hypothesis as well.

Let's go far back to start with (please bear with the longish answer. Couldn't help it as story of India is so long!) -

5,000 year ago - Indus Valley Civilization - Religion in the Indus valley seems to have involved temple rituals and ritual bathing in the 'great bath' found at Mohenjo-Daro. There is some evidence of animal sacrifice at Kalibangan and there are archaeological finds indicating sculpture from that period (for instance the famous statue of dancing girl). The Pashupati Seal has been identified by few scholars as an early prototype of the Hindu god Shiva (or, his Vedic predecessor, Rudra). After the collapse of the Indus Valley civilization around 1900 BC, there is little record of sculpture for next 1,500 years until the Buddhist era. There's one hypothesis that links Dravidian culture to the Indus Valley Civilization but no evidence as such.

1500BC - 500BC - Vedic Period - Few key things stand out from this period like animal sacrifice, Fire worship, and rituals involving recitation of Vedas. Vedi is the term for "sacrificial altar" that were an elevated enclosure, generally strewed with Kusha grass, and having receptacles for the sacrificial fire.

Its possible that the temple structures in Vedic age were not made of stones or bricks, which came much later. They were possibly made of clay with thatched roofs made of straw or leaves which have not survived till modern times.

Atharvaveda was last to be written of the 4 Vedas around 700 BC. One of earliest mentions of Sanskrit word Silpa or Silpin is in Atharvaveda which refers to the craftsmen and builders (of temples).

322BC - 185 BC - Mauryan Empire - Ashoka being a strong promoter of Buddhism, it was at peak during his reign. Stupas like Sanchi, Bharhut, Dhamek originated in this period (and were extended further by later kings).

The 4th century BC text, Astadhyayi mentions male deity arcas (images/idols) of Agni, Indra, Varuna, Rudra, Mrda, Pusa, Surya, Soma being worshipped, as well as the worship of arcas of female goddesses such as Indrani, Varunani, Usa, Bhavani, Prthivi and Vrsakapayi.

Kautilya’s Arthasastra from 3rd Century BC describes a city of temples, each enshrining various Vedic and Puranic deities.

Note that we don't have any surviving remains of palace of Chandragupta Maurya or Ashoka. Possibly because they were constructed of wood and other materials that did not survive 2,000 years. Temples of this period may have gone through the same challenge.

100 AD onward - A new type of worship known as Bhakti or devotional Hinduism spread across the Indian sub-continent, and the old Vedic gods were replaced in importance by deities like Shiva, Vishnu, Krishna, Brahma, and Devi. These gods would become the central figures of Hinduism and their worship required temples where the devoted could offer their thanks and reveal their hopes for a better life.

From the 1st century BC through 3rd Century AD, the evidence and details about ancient temples increases. The ancient literature refers to these temples as Pasada (or Prasada), stana, mahasthana, devalaya, devagrha, devakula, devakulika, ayatana and harmya.

320 AD - 550 AD - Gupta Empire - Also known as Golden Age of India.

GUPTA CAVE SHRINES

The earliest examples of religious architecture were cave-temples. Inside the shrine, ritual sculptures were placed such as a Shiva linga (phallus) and the walls were richly decorated with more carvings showing scenes from mythology. Notable examples are found at Udayagiri in Madhya Pradesh where one cave bears a date mark of 401 AD.

FREE STANDING TEMPLES

Not satisfied with caves the Guptas were the first dynasty to build permanent free-standing Hindu temples and so they began a long tradition of Indian temple architecture. The first materials used were wood and terracotta, but architects gradually moved on to brick and stone, especially sandstone, granite, schist, and marble. No mortar was used in the older temples and so precise cutting of dressed stones was required. Early free-standing temples survive at Deogarh and include the 6th century Dasavatara temple dedicated to Vishnu (picture at top of answer).

Subsequently, this growth phase of Hindu temples charts its rise and fall alongside the fate of the various dynasties that reigned India during the period majorly contributing and influencing the building of temples, especially in South India.

In South India, this period is credited with some of the earliest stone temples of the region, with Chalukya temples dated to be from the 5th century

6th century - 13th century - Glorious temples were constructed in what is now Karnataka, Orissa, Tamil Nadu, Madhya Pradesh, Maharashtra, Rajasthan, West Bengal, Uttar Pradesh.

In north India, Brihat-samhita by Varāhamihira is the widely cited ancient Sanskrit manual from 6th century describing the design and construction of Nagara style of Hindu temples.

By the 6th century AD, several Sanskrit manuals for constructing palatial temples were in circulation in India. These manuals describe the education, characteristics of good artists and architects. The general education of a Hindu Shilpin in ancient India included Lekha or Lipi (alphabet, reading and writing), Rupa (drawing and geometry), Ganana (arithmetic).

Manasara, a text of South Indian origin, estimated to be in circulation by the 7th century AD, is a guidebook on South Indian temple design and construction

Isanasivagurudeva paddhati is another Sanskrit text from the 9th century describing the art of temple building in India in south and central India

The Silpa Prakasa of Odisha, authored by Ramacandra Bhattaraka Kaulacara sometime in ninth or tenth century CE, is another Sanskrit treatise on Temple Architecture

In South India, the Pallavas (600 - 900 AD) sponsored the building of the rock-cut chariot-shaped temples of Mahabalipuram, including the famous shore temple, the Kailashnath and Vaikuntha Perumal temples in Kanchipuram in southern India. They were followed by Cholas, Pandyas, Vijayanagar kings. The Cave Temples of Badami, the Virupaksha temple at Pattadakal, the Durga Temple at Aihole and the Kailasanatha temple at Ellora are standing examples of the grandeur of this era. Other important architectural marvels of this period are the sculptures of Elephanta Caves and the Kashivishvanatha temple.

Buddhism/Buddhist Stupas Vs Hinduism/Hindu Temples - Buddhism and Hinduism share a complex relationship from 6th century BC (when Buddhism came to being) till 12th century AD (when Buddhism diminished in India). There are several instances of Brahminism being at odds with Buddhism and at the same time there are instances of both being in harmony and in fact evolving out of each other.

In terms of Architecture, the simple shape of the early stupa may have evolved into the complex superstructure of later Hindu temples.

Ellora caves built between 5th and 10th century, are one such instance of harmony between the two. The 34 "caves" include 12 Buddhist (caves 1–12), 17 Hindu (caves 13–29) and 5 Jain (caves 30–34) caves, built in proximity.

The Ajanta Caves which date from the 2nd century BC to about 650 AD, on the other hand are exclusively Buddhist.

Greek Influence -

Greco-Buddhist art is the cultural mixture of Classical Greek culture and Buddhism, which developed over a period of close to 1000 years in Central Asia, between the conquests of Alexander the Great in the 4th century BC, and the Islamic conquests of the 7th century AD.

The Kushan dynasty (Kanishka) ruling out of Peshawar and with close links with Mathura promoted Greco-Buddhist art.

This is the period when we see the first representations of the Buddha in human form. It's possible that the idols of Hindu deities seen later are a manifestation of this form of sculpture. The Hindu art of Gupta empire was certainly inspired by Greco-Buddhist art.

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First of all, it depends on what you classify and call as "Hinduism". If you consider only Vedic tradition as synonymous with Hinduism, then the dates would be around 5th century. But long before this time, there was a strong widely spread non-Vedic and non-agamic sect of religion in Tamil Nadu. There were other parts of India with such religions too.

These deities are known as 'Kaval dheivam' (guardian deity) or 'Kula dheivam' (family gods) in Tamil Nadu/Andhra Pradesh. The deities may be a representation of local folklore depicting a local hero or a village protector. A lot of these deities have been sanskritised and amalgamated into the later Vedic religion. One of the biggest such example is 'Seyyon' or Kumara kadavul (aka Karthikeyan), who was mentioned in early texts as the God of the Tamil diaspora. Also Kotravai (aka Durga) and Maayon (Vishnu) have been amalgamated in to Vedic gods. The rural areas of TN and Andhra Pradesh still have these temples and unique rituals. Some of their deities are still independent of the Vedic deities. Ayyanar, Kali, Madurai Veeran, Karupannan Swamy, Sudalaimaadan, Muniyandi are few of the famous deities that play an important part in rural spiritualism in South India.

Statue of Karuppana Swamy

These deities have always had temples dedicated to them, usually built in the border of the villages. Note that temple worship is not an essential part of Vedic religion, whereas the temple tradition have always been a part of these Dravidian religions.

Here is a excavated brick temple of Karthikeya in Tamilnadu, which dates back to 3rd Century BC. Note that the temple has features of temple built with agamic tradition, with a bali-peetam, dwajasthambam, sanctum sanctorum, etc.

So going by this history, and with the assumption that this branch of tradition is called Hinduism too, the temple tradition must have started for these village deities - long before Buddhist and Jain temples existed in India.

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With constrution of stupas construction of temples also started as free standing structures .temple architecture is not much old as it seems .the primary temples were based on mythical stories of Puranas, Basic structure of temple included -
1-Garbhagriha- cave like sanctum in which main deity is place after ritual of pran prathistha
2-Mandapa- is entrance of temple
3-shikhar- it is like tower ans from 5 AD construction of this tower started known as shikher in north india and vimana in south india
4-Vahan-it was deity's vehicle or mount which was placed along with dhwaja axially sanctum
among them Dashavatar temple of Uttar Pradesh,Nachnakuthara and udaygiri in vidisha (MP). all of f these temple are earliest structures consisting of verandah, a hall and a shrine in rear
the rock cut temple reached its zenith in gupta era and marked the begining of free style temple architecture.most of temples in gupta era was built with avtaras if vishnu , lingum and goddes.
some of oldest temples in india are -
1)Dashavtar temple in LALITPUR district of uttar pradesh which was discovered by captain charles strehan and named by Alexander canninghum ,the main shrine is bult in rectangular plinth with four smaller subsidiary shrines at four corners hence its name is panchaytana,
interesting part of this temple is that it is west facing which is less common as most of temple in modern times is east or north facing .this is also known as sarvatobhadra temple .
2)BHITARGAON TEMPLE- in kanpur district of uttar pradesh is oldest remaining Hindu temple was built in Gupta era in 6th century
where as if we talk about south Indian temples then they mainly started in 7th and 8th century with chalukyas of badami .they built nearly 100 structures both rock cut and free style in his karnata dravida architecture. this was mainly mix of northern NAGRA style
and southern DRAVIDA style
PS- I have tried to give correct info ..but if i have written any thing wrong then please correct me and also tell me additional info about temples in india