Mmmm, it has nothing to do with metal getting hot on "helmets". First, we out west woodsy folks don't wear "helmets" we wear hardhats. The metal ones were worn for eons on the fireline. I've never heard of them getting too hot. We started to melt a radio on a belt one time digging line, but nobody got a fried head from the aluminum MacTs. In fact, I think they are cooler to wear in hot weather. They can get noisy in the rain and brush.
One reason given for the switch to plastic hard hats was electricity. Working near powerlines was what we heard. You might get zapped. That sounded kind of far fetched, but it was what the rumor was at that time. Then the order came down in the Forest Service that everyone was to switch to plastic, fire crews and others. We timber folks picked nice colors and went on with our work. I came here and we were told we must have the heavy, made for fire plastic hats and I can't think of the brand name but many of you will. They are heavy. Everybody had to have that kind and the reason given was questionable--so "anybody going to a fire could grab one and have the proper kind". Not that my hardhat was ever grabbable. This decision cost quite a bit of budget money. Oh well...and I hated wearing the heavy thing doing timber work.
Since then I've heard that the reason the switch to plastic was ordered was dents and damage to aluminum hardhats. You tend to throw them in the rig after work, or I do, and sometimes fling them in the pickup bed with the other gear. They get dinged up.
A dented hardhat is not a safe hardhat. I've seen some terribly dented ones worn on logging jobs, and ones where the "clever" wearer has deliberately bent the brim up. If a safety inspector showed up, those folks would be fined, but there aren't many safety inspectors.
That's it. And I have never worn a skull cap brand so I have strayed from the original question.