Although the aim is good I doubt body temperature and a hot sun will permit the cool pleasure to last very long.

To make it last a useful time, you would have to make the gel pack pretty big and heavy. That might be a bit of a disadvantage in a sporting event. If you had an ice chest full of spares, I suppose you could just change them out periodically. There are chemicals that can generate cold so to speak via chemical reaction. I don't know if that would work better or not. It would certainly be cheaper to have reusable gel packs than one use only chem packs.

A variation of the "Cobber" is what you probably want. they're designed to go around your neck. They are made of cotton and stuffed with a substance that absorbs water when you run it under a faucet. As the water evaporates it keeps your neck cool, which is supposed to cool your head.

I've seen motorcyclists wearing them. (never tried it myself tho'.)

There are many ways this could be done. The way you mentioned is one. Also you could have evaporative cooling, so when the wind blows the wet hat cools down. Also a new technology is being researched called Active Building Envelopes(ABE) that could be applied to a hat for solar powered thermo-electric cooling.