Biological methods to reduce internal body temperature:
* Increase surface area of skin, via cracks, and folds (mentioned elsewhere) like an elephant.
* Use extrusions of the skin to increase surface area, and use them to pump blood through to cool blood. Elephants have this with their ears, but some elephants don't need this, so have smaller ears. Lets compare the african and asian elephants.
* Use internal openings, wet from internal moisture, to cool off. Many, if not most mammals, use this method of "Wet" cooling off in dry weather with their tongues, and the inside of their mouths.
* Even plants attempt to cool off with this method, though it does appear like sweating, it is more so that holes inside leaves open, rather than water is excreted out.
* Use externally wet methods of cooling off, ie sweat glands what humans do. Requires mostly exposed skin for this to work.
* Light coloration can be used to reflect radiated energy instead of absorbing it and turning it into heat.
* With mammals near the equator, usually this coloration applies to the fur, often what happens is this coloration is tied into the environment, so not completely white. The skin of the animal however is still very dark. The production of melanin is used to block UV radiation, preventing skin cancers. You'll find many animals with dark noses and even tongues because of this melanin issue, even in hot environments. It's common knowledge that this applies to humans, but it also applies to dogs like the Xolo breed dog of Mexico, which actually tan like we do. This isn't so much a heat adaptation, but an equatorial adaptation
* Lower metobolic rate, to decrease internally produced body temperature. The metobolic rate of elephants for example, is much lower than that of small animals like mice:
* Use ectothermy, to derive your temperature mostly from the outside world. Reptiles do this, and have created some truly massive predators that weren't dinosaurs, but traditional basal reptiles and basal archosaurs (snakes, turtles, crocadilimoprhs) were able to achieve massive sizes. Ectothermy, cold-blooded ness, is where body temperature is mostly determined by the environment (though, even ectotherms, at very large sizes, will start having internal dynamics with increasing body temperature with out adaptation). 671
During the period directly after the fall of the dinosaurs (paleocene), the largest animals on earth weren't birds, or mammals, they were reptiles, reptiles at sizes bigger than they had ever been even during the Mesozoic (reign of the dinosaurs). The temperature of the earth was very hot, hotter at some points than it was during the age of the dinosaurs, and certainly hotter than today. This, in combination with niches opening up, enabled massive reptiles to prosper, like titanaboa, Carbonemys, and various extremely large crocodilians like Euthecodon.
* Use of Mesothermy. Mesothermy is basically anything in-between ecto and endothermy. One hypothesis for why Sauropods were able to get so big is their use of mesothermy. While many dinosaurs were likely endotherms, albeit with possibly lower metabolisms like the African Elephant, at the sizes Sauropods reached, they would have cooked themselves. However, despite being so large, Sauropod babies were extremely small comparatively, "no larger than geese", their parents were "10000 times heavier" than they were. At those sizes, baby sauropods would likely not have had ectothermy, as they'd otherwise be relegated to the same niches as similar sized reptiles, though even more vulnerable. What's more, is that dinosaur babies grew extremely fast. They couldn't afford to be slow lumbering beasts, these babies had to eat a lot and eat fast. Hypothesis have been thrown out that the smaller babies had different metabolic characteristics than adults, and that adults metabolism wouldn't have matched either ectothermy or endothermy. The idea that traditionally ecto/endo/thermic animals could achieve something different than strict cold/warm bloodedness or transition has been seen in naked mole rats, which have transitioned to a mostly ectothermic lifestyle, and some monitor lizards which have developed a somewhat ectothermic+ strategy.
Added Thoughts
A few things I thought off the top of my head that I didn't exactly feel like would fit here, but I think are worth mentioning.
* While sort of behavioral, licking is used to cool off, particularly with mammals. Dogs, Cats, Kangaroos, they will lick themselves in order to cool off, though I think this may go against the spirit of the question, not only due to the behavioral nature, but also because this is a temporary measure against heat. 671