I think you might be thinking about this slightly wrong.
Cooling can be thought of as simply moving heat from one place to another. You take heat from one place and move it to a different one. If the first place is hotter than the second then 'cooling' happens all by itself as the temperatures equalise. If you want them to equalise faster or you want to move temperature from a hot place to a hotter place then that is when you have to use energy. All a refrigerator does is moves heat from the inside of the refrigerator to the outside (the pipes on the back are a radiator to get rid of the heat from inside).
So no, there exists no material that will 'just cool' all on it's own. That would be a violation of the laws of thermodynamics. You can have materials with a high heat capacity that you can dump heat into (Radovan Garabík's answer is a pretty good example of that!), but first they'll have to be cooled down, or start off cold. The bottom of the ocean is a good one for this: water is at it's densest around 4 degrees, so the bottom of the ocean is always around that temperature. You can pump heat into the ocean all day long, though it might have some unintended consequences. Eventually though your heatsinks will reach the same temperature as the thing you're trying to cool, and you'll have to spend more and more energy to move the heat from A to B.
But this brings us to endothermic reactions. If you have two materials which, when brought together, require energy in order to bond then they will suck heat out of the air nearby. Again though you run into a problem: These reactions will eventually stop, and you'll either need to use energy to break the chemicals apart again (thus releasing the heat) or introduce more of the chemicals you used in the first place. The exact whys and wherefores of how you can do this and what chemicals might be good is more chemistry than I'm comfortable professing to know, but it might help, especially if you have a way of shunting the result off world and bringing in more of the raw chemicals from somewhere else.
The third option is using the heat to extract usable energy, then shoving that energy somewhere else. Remember I said the ocean was a good heatsink? Well, it is. Turns out if you pump it up to ground level and equalise it's temperature with the surface water you can pull out usable energy, essentially using the entire ocean as a refrigerator, but in reverse. This actually works with wind as well, as winds are driven by temperature differentials. What you do with that energy is then up to you, but remember that if you use it on Earth all you're doing is (again) moving the heat around. Try firing lasers into space. If you're clever enough you might be able to balance the incoming and outgoing radiation and halt global warming. Or maybe just build a really good renewable power grid.
Heck, lets get crazy.
So, the problem is that you need somewhere to put the heat that isn't 'here'. So, first lets say that there are multiple universes, and that they're accessible. Second, lets say that we know some of these universes are much colder than ours, and some are warmer. Third, lets posit the existence of a specific crystalline matrix that will act as a 'bridge' to a universe colder than ours. Any crystals that can be convinced to grow into this 'Unobtanium matrix' will then act as a (potentially infinite) heatsink, into which we can pump heat with reckless abandon.
Which may piss off the frost giants on the other side...