For most of our 25 years in this house we didn't have air conditioning on the first floor where the dogs, bulldogs, lived. Bulldogs are especially sensitive to heat and they get heat stroke easily and sometimes die. So it was always something that I watched. We live in St. Louis with hot, humid summers.
But we
1) DID have air conditioning upstairs, so some of that cooled air fell to the first floor (though didn't always reach the dogs in their dog room--our breakfast room)
2) have a Victorian town house with high ceilings and few windows--it is built to be cool
3) turned a fan on the dog room when it was hot, but made sure they had a way to get out of the fan wind--that can be drying to their skin and can be stressful. I like to set an oscillating fan on the table and that makes air move through the room, but down at the dog's level, it's not blowing in his/her face.
In the two final years of our elderly bulldog's life, DH put in a window air conditioner for her. It was one that had come with a piece of real estate, and it was huge. I remember him lugging that huge, ancient thing home and hoisting it up into the window for her, a big job for 1 person, but he loved her. And it really eased her final summers.
About 3 years ago we installed air conditioning for the first floor of our house, so now the dogs are fine in these summers.