For a long time I had a traditional black-velvet covered riding helmet. This cooked my brain in the Australian sun (and perhaps explains a lot of things
). You wouldn't believe what a difference it made when I bought a helmet that was white, shiny and well-ventilated!
Also, no long boots, no plasticky riding pants that feel like a sauna to wear - there's cotton / elastane, in riding pants as well as jeans (the maroon things in the photo are actually $5 cotton / elastane op-shop jeans that are stretchy and really comfortable to ride in and have no bling or studs to scratch my saddle), a nice cool breathable top - some modern synthetic materials for athletic clothing are wonderfully cool and comfortable, even for me who dies a slow death in anything with polyester in it. My new cycling jersey is amazing for keeping cool, for instance, and similar material garments are available for endurance riding - without the Tour de France
Bamboo socks make all the difference in the world too... try those sometime. I've found nothing better for hot weather, and they're really cool and comfortable - even though quite thick!
In the summers when I was a teenager many moons ago and living on the West Coast of Western Australia, it got so hot for weeks on end in summer that, apart from riding early in the mornings or in late afternoons, I'd sometimes give up and ride barefoot, in shorts and singlet, without a helmet, and without a saddle on the horse. We even did some endurance training like that, running up the sand hills dodging kangaroos, but you know what teenagers are like...
In really hot weather, you can also spray yourself head to foot with a garden hose on the fine spray setting in your riding gear before you get on your horse. Evaporative cooling is lovely... In really hot weather, we often sprayed the horses as well before work...