Moving air, shade and water is what is key.
But you are right, different places, different heat, different humidity all make a difference.
I would never even think of a fan, the wind in SD is pretty steady, sometimes hard enough to move a log chain!
Check your shade at different times of the day.
Shade, space, and enough water. I really would not put ice in it, it can cause shock. Just a lot of water. Sometimes I wet the ground, I have mostly sand, and they like to dig and mud bath in that.
Mrs K