Get it cool, quick!
Depending on which season and species I'm hunting, the temperatures can vary significantly, from below freezing to mid 80's. I've long been told by a very wise man (Dad) that it's important to get the carcass cooled as quickly as possible. After the kill, immediately gut the animal. The hide is only left on long enough to get the animal back to camp as clean as possible, and then removed. If it's cool enough, we hang the carcass in the shade, typically overnight when Colorado temps drop significantly. Then it's time to pack up the critter and get it into refrigeration, and then only as long as it takes to get the animal processed, either by a pro or by yourself. As mentioned above, heat is the enemy, and if a nice thick elk or deer hide keeps the animal warm when they're alive, it will certainly keep them warm when they're dead.
Good luck!
Glen