Some hats have mesh tops, so you still get some airflow on your crown. I'm pretty sure the sunhat i got for gardening does, but i've not worn it in a while.
You could modify a Mexican sombrero with a high brim to hold water, and then if you get hot you pull a cord to uncover a drain hole that drips water on your face. The more i write the less good this idea seems.
Other options are carrying a parasol/umbrella, living in a giant fishbowl on a wagon so you're always swimming, only ever being underground (caves are very cool), or wearing much sunscreen.
Wizard/witch hats are neat because you can use the top for storage while the brim works as a brim. You could just keep an insulated thermos of ice in your hat, and take some out to eat or drop down the back of your shirt whenever you get too hot.
I guess you could build a portable awning that attaches to your shoulders, but you might have trouble getting thru doorways.
