I think you need to identify what makes the headphone hot for you. Is the driver actually creating heat or is it just the ear cup against your head? If that is the case then you might try some IEMs.
The only other thing that comes to mind is something like the AKG k1000, which are discontinued and were well outside your budget. This didn't cup the ear so probably would have been cooler.
AKG K1000 Headphones Review By Alvin Gold
Unfortunately I'm not aware of any other headphone that sits on the head like that. The problem with a design like that is it's obviously not going to offer anything in the way of isolation.
On ear will have less contact with your head, so may be cooler. However, on ear also means it tends to press on the ear. I have Grado sr60's but I can't wear them for longer than an hour. I've read you can swap out the cushions for the LCush for more comfort but i never did.