Tucked away in the woods of Hopkinton is an abandoned children's amusement park with an enchanted theme. If you know where to look, it's a neat place to explore and take photos and with any luck, either the woods or the creepy chills will help keep you cooler than you might be otherwise.
It might not be super easy to stay cool depending on when you leave, but the Providence Preservation Society offers leashed dog-friendly walking tours, as does the Providence Ghost Tour, which is perfect for hot days because they only tour at night, which means it will be considerably cooler.
If you're in or near Providence, stop into Books on the Square. They're dog-friendly and even regularly put out a bowl of cold water for dogs stuck in the summer heat. While they lap up some refreshments, you can lap up some new reading material.