When was the first Hindu temple built in India? No

 No one can tell, when the first Hindu temple built in India. But the three literary gatherings around Madurai traced back their origin to around 4400 years ago mentioned there existed brick shrines of Mayon (Vishnu), Seyon (Karthik), Kotravai (Durga) etc and temples of local hero's.

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Legends: The Tamil tradition and legends link it to three literary gatherings around Madurai and Kapāṭapuram (Pandyan capitals): the first over 4,440 years, the second over 3,700 years, and the third over 1,850 years before the start of the common era.

1. The early Dravidian religion constituted a non-vedic form of Hinduism in that they were either historically or are at present agamic.
2. Ancient Tamil grammatical works Tholkappiyam, the ten anthologies, Paththuppattu, the eight anthologies, ettuthokai, sheds light on early ancient Dravidian religion.Seyyon (Murugan or Karthik), was glorified as, the red god seated on the blue peacock, who is ever young and resplendent, as the favored god of the Tamils.
3. The Sangam landscape was classified into five categories, based on the mood, the season and the land. Tholkappiyam, mentions that each of these landscapes had an associated deity such as Seyyon (Murugan or Karthik) in Kurinji-the hills, Thirumaal (Vishnu) in Mullai-the forests, and Wanji - Ko (Indran -the god of Thunderstorm) in Marutham-the plains, and Varuna (the Water God) in the Neithal-the coasts and the seas and Kotravai (Mother Goddess) in Paalai - the desert. (instead of wasteland, it is also taken as a mixture of Kurinji and Mullai under the heat of burning sun)
4. Many local deities, called ayyanars, karuppasamy, pechiamman, believed as the kul dev (i)s are local heroes who protect the village or their family from harm. Their worship often centres around nadukkal, stones erected in memory of heroes. This form of worship is mentioned frequently in classical literature and appears to be the surviving remnants of an ancient Tamil culture.