From the Mythology Dictionary:
The hare on the moon is associated with Sakka or Sakha (Son of Skanda who is war god most associated with Orissa and Bengal and Tamil Nadu, see Integration of Dravidian Murugan and Aryan Skanda gods, see also the British Museum’s “Stone Figure of War-god Skanda”, of Hindu Origin: A god, an aspect of Indra. Ruler of Tavatimsa.
A hare, an otter, a monkey and a jackal, living together in the forest, all agreed to give food for the needy. The hare could find no food and offered his own body. When Sakka heard this, he set out to test the hare who willingly offered himself to the god who then built a fire to cook himself a meal. The hare, a previous incarnation of Buddha, threw itself into the flames and was taken up to the heavens. An outline of the hare, which many say they can see on the moon, was drawn there by Sakka. Occasionally called Sakka,Indra, Indra, Aptya, Bhatara Guru, Divaspati,Gadhi, Is(h)vara, Is(h)wara), Mahendra,Meghavahana, Prajapati, Purandara, Sa-yoni,Samraj, Sarasraksha, Sayoni, Shachipati, Shakra,Svargapati, Trita, Vajrapani, Vajri, Vasava,Vishvakarma, Vritrahan, Vrtraghna, Buddhist Amoghasiddhi, SakraBurmese Hkun-sa-ya,Cambodian Prah En, Chinese Ti-shih, Wei-t’o, Yin-to-lo, Greek Zeus, Japanese Shaka,TaishakuPersian Verethragna, Tamil Venda,Pajanya, Trita Aptya, Sakkan, Sakkan, Amakandu,Sakka, Hittite, Sumukan, Sumuqan or Sumerian Lahar.
Source: Mythology Dictionary