I have a running hat (actually two identical ones) from a company I don't like, but it works perfectly for several reasons:
1. Blocks the sun in winter, keeps the wind off my scalp in the winter.
2. Sweat drips from the brim instead of into my eyes when it's hot, rain does the same in the rain.
3. The hat is white, which is good for cooling, except the bottom of the brim is black, which is good for glare reduction.
4. The hat is machine washable, I think it has a piece of rubber in the brim that maintains the shape. That means I can just throw them both in the laundry to wash them regularly.
5. It looks like a normal hat, people don't give me weird looks or anything. I think women actually tend to smile at me more when I'm wearing it.