I wear a lightweight skull cap. It has a front absorbent strip built in, but the whole thing, I thnk, soaks up and/or facilitates evaporation and therefore keeps the sweat from running down my face. I have several, of various weights, with some being extremely light weight. But none of them make my head feel hotter.
Assuming you use sunscreen on your face, maybe try a different brand. We use a specific variation for "faces" and it definitely doesn't sting as much (not at all) when I sweat compared to when I use regular sunscreen on my face. My stinging eyes is almost exclusively from the sunscreen being washed into my eyes from sweat.