You should be fine with him in a cool room, just dont take him str8 out of the cool room and throw him on the pasture in this heat, wean him to it gradually, but cattle actually handle the temp changes a lot better than most people give em credit for.
My only concern would be more from a movement standpoint. Probably fine for baby bulls, but I would think an older (yearling +) one needs more room to roam on a daily basis to keep from getting stiff.
Well I dont think his bull is going to be living in the cool room...should follow the same routine as any show calf put in a cooler, in during the day turned out at night...its not going to make a bull any more stiff than it would a steer or heifer.