Bigbore,
You are 100% correct. Heat is a major concern when hunting in Texas in the summer. However, if you are after a really good axis buck then the summer is the best time to hunt them. Unfortunately other exotics are best to hunt during the fall-winter-spring time. During the summer all of the other deer species are growing their horns just like our whitetail, this includes fallow, sika, red stag, barisingha, etc, etc. Most of the blackbucks are in their reddish summer color phase then and almost all of the rams have lost their heavy winter coats and don't look as good without their heavy beards. Even the aoudads have shed their winter coat. A bright note is that the African hoofstock are pretty much ok to hunt year round, if you can afford them. Feral hogs can be hunted in the summer but I prefer to hunt them in the fall after the first cold front, the meat just seems to taste better then.
Besides heat you will have to worry about snakes and bugs. The hot weather seems to bring out the snakes, tick, mosquitoes and other biting critters. AS it will be hot you more than likely will want to hunt in shorts and a short sleeved shirt so an odorless insect repellant will be needed. (Yes most exotics can smell as good as our whitetails, so unless you hunt them from a moving vehicle, you will still need to exercise scent control.) And the downside to hunting in shorts is that my snake chaps tend to rub my calves.
But if the heat and bugs dont bother you then hunting during the summer can be a lot of fun. You get to see the country at a different time of year than hunters are used to and there are very few people hunting to interfere with your activities. And except for axis deer, most lodges seem to have cheaper prices during the summer off season. Good luck.