Thousands of people tried to force their way into the temple when its gates opened at daybreak in Deogarh, a town in Jharkhand state, said a police officer.
At least 11 pilgrims, including eight women, died and another 24 were injured, he said. The injured were being treated at a local hospital.
He said tens of thousands of people were attending a month-long festival at the temple of Shiva, the destroyer, one of the main deities of Hinduism.
The town is nearly 160 miles south of Patna, the capital of Bihar state.
Stampedes are fairly common during Indian religious festivals, where large crowds gather in small areas with few safety or crowd management measures.
In July, 27 people died when tens of thousands of pilgrims taking part in a Hindu religious bathing festival triggered a massive stampede on a riverbank in southern India’s Andhra Pradesh state.
In October 2013, a stampede in Madhya Pradesh state in central India killed more than 110 people, mostly women and children.