If you're near Asheville, check out the Annual Doggy Dip which takes place at the Recreation Park Pool. They have designated hours depending on your dog's weight and size, so if your patience for the heat is growing thin, go and take your dog for a swim!
Not every city will have a great escape from the heat but Wilmington does in the form of a nearly 2-mile boardwalk. There are numerous restaurants with umbrellas to provide cool drinks and shade for both of you and even boasts some dog-friendly businesses you can go inside to catch some AC relief.
It won't be cheap, but renting a beachfront cabin means you get to enjoy the cool breeze off the ocean with windows open and a ton of shade. You can even draw a cool bath if you don't want to head to the shoreline to enjoy the Atlantic itself.