Is my cat's fast breathing normal in hot weather? How to keep cool?

Fast breathing can be a normal way for you pet to cool himself. If the house is tolerable for you then it is likely tolerable for your pet. However, labored breathing and open mouth breathing in a cat usually indicate a more serious problem concerning the lungs and a vet visit right away is highly recommended.
However, good ways to keep a pet cool is to set up fans in the home (if A/C is not in place), therefore when the pet gets hot he can just sit in front of the fan to cool off. Floor fans can be purchased from your local hardware store or department store. Keeping water available at all times is also helpful. Most cats don't like blankets when it is above normal temperature, so this is not uncommon. Best wishes taking care of your pet Hans and I hope everything turns out ok for him. Take care.