Millonas wrote: "The top of it is essentially a thin skull cap. Forget about UV, what about IR. It great for what it blocks, but anything it actually adsorbs gets transferred directly to my noggin, with not too much air to let it escape. Still for high mountains where temp is not necessarily high, but where light is very intense is is probably perfect."
Yup. A ballcap is an insulating piece, just like a very thin fleece hoodie. If the problem is UV rather than excess heat-minus-windchill, all well and good -- I've never walked above 4000 feet, but I can imagine it would be good for that.
If you're just hot and sweaty, you want shade and ventilation. Get a grimmed hat with several square inches of mesh, or better still a reflective umbrella if it's practical (wind etc).
And if you just want to keep sun out of your eyes, get a strap on sun visor.
[Fun fact for desert rats: in Britain we have fifty times as many deaths invloving a lack of UV as a surplus.]