Its a weird restriction...mountains would be relatively cooler which you dont want to go,beaches anyways u dont want to go to. In that case you can visit Udaipur, Kumbhalgarh, Chhitorgarh in Rajasthan
Visit any Hill station in June, it's not cold there in that season. Rajasthan or any northern Indian will be very hot, and South will be wet bcoz of rains.
Please tell about the starting Destination. it would be better
you can go to Madhya Pradesh .
Rishikesh ( not a proper hill station but you won't be able to do rafting in June july )