Do there exist objects that absorb energy only (and do not give it away)? Do there exist objects that absorb energy only (and do not give it away)? Magnets have been mentioned in this post, do we know more examples? Extract energy from magnets
And how is this capability of giving away energy or absorbing energy clustered and theoretically modeled?
 A: Such an object would almost certainly violate the known laws of physics. There are a couple of ways to look at this.
At the quantum mechanical level, for a system to absorb energy, there must be some sort of interaction. All known interactions can be reversed: if the system can absorb energy via that interaction, it can also emit energy via that interaction. And if it absorbs energy, it enters an excited state. That excited state will have a finite lifetime. so the system will eventually emit that absorbed energy.
At the level of thermodynamics, any system with finite temperature will emit energy in a black body-like spectrum. If it can absorb energy, it will have a finite temperature. In fact, we have never seen a system with exactly zero temperature.
One possible exception, as other answers indicate, is a black hole. Material goes into a black hole, but nothing comes out. But there is a loophole: black holes are believed to emit Hawking radiation. This is radiation that is created just outside the event horizon of the black hole. Some of that radiation is believed to escape from the black hole entirely. Hawking radiation has not yet been observed, so it is still technically possible that black holes can absorb energy but not emit it.
A: I expect such an object would violate the laws of thermodynamics so I don't think it's likely. An energy exchange with this body would occur at 100 % efficiency which seems very fishy to me -- dangerous waters.
Some comments mentioned black holes, but even those have to "leak" energy due to the laws of thermodynamics (search for Hawking radiation on Google). It's kind of like the speed of light -- you can approach it with arbitrary precision, but you cannot physically travel at the speed that exactly matches the speed of light in the vacuum. However, black holes are indeed the closest thing to what you described, so maybe this would be an interesting read: http://arxiv.org/abs/hep-th/0409024 (or at least parts of it).
