How do animals lose heat?
There are a number of ways, which are briefly summarized here. I also recommend reading this nice Scitable article on thermoregulation. It is not too technical. Felines do pant when they get hot. Horses do sweat. Jackrabbits can enlarge the blood vessels in their large ears to eliminate excess heat. Bird use a process called gular flapping, which is similar to panting of dogs and cats.  Animals can move into shaded areas.  The possible mechanisms are many.