In Uttar Pradesh state, 18 people died as summer temperatures climbed above 42C, said Rajkumar Singh, a police official in Lucknow, the state capital.
Ten people have died from sunstroke in the eastern coastal state of Orissa, said Manmohan Samal, a state minister.
Angry protests have taken place against the power cuts, and three people were arrested after two government vehicles were set on fire in Uttar Pradesh.
Power cuts in the capital, Delhi, have lasted up to 10 hours and in rural areas of the country people are receiving as little as two hours electricity a day.
City officials have called on residents of the capital to use just one air conditioner as temperatures push into the mid-40s celsius.
"You know how crazy the idea of turning off air-conditioners is - how hot it is here? You could roast a leg of lamb in this heat," said actor Roshan Seth, a Delhi resident and urban activist.
The city government had ordered shops to close at 7:30pm in an attempt to conserve electricity but was forced to withdraw them after traders protested.
Rampant power theft and India's underdeveloped power infrastructure have been blamed for the power shortages.
In the past 10 years demand for power has increased by 12%, whereas power generation has increased by just 5.5%.
A $2.9 billion power plant is set to open in Maharashta, India's wealthiest state – but 47 planned plants in the country are behind schedule.