Like others I use caps. They have a perfect sized peak to keep the sun off your head & out of your eyes, the cap itself absorbs sweat, and as you ride along the relative wind evaporates the sweat and keeps you coool (copyright Hacienda71).
On a hot day, dip the cap in a fountain or run it under a tap, then pop it back on your head, instant cooling till it dries off - then repeat.
With the peak upturned it is a relatively safe place for riding glasses to perch, and off the bike you can hold keys, phone, wallet and other small items in it as if it were a small purse, while you organise yourself post ride.