Under General Relativity (GR) alone, a Black Hole's (BH's) event horizon is a point of no return -- anything that passes through the event horizon is lost and gone forever, and nothing comes out. Hence, under GR alone, BHs are utterly black and don't have a temperature at all.
This is why the absorption of radiation (or anything else) by a BH doesn't raise its temperature -- it just gets swallowed up and lost. (It's mass, angular momentum and charge do remain, but that's all -- see the No Hair Theorem.)
(Note: The accretion disk that surrounds a BH can be very hot indeed, but that's another thing entirely.)
Stephen Hawking discovered that applying quantum mechanics to BHs showed that BHs would emit a random spray of radiation, and that that radiation was precisely what a black body would emit -- black body radiation. This is called Hawking radiation.
Black body radiation is simply the thermal emission of a perfect absorber of radiation, and leads to the inescapable conclusion that a BH does have a non-zero temperature. Interestingly, Hawking's analysis showed that the effective temperature of the BH is inversely proportional to its mass and that solar-mass BHs (which are the smallest for which we have actual evidence) would have a temperature of about 0.00000006 K. Kinda cold, but still not zero.
Note that, unintuitively, a solar mass BH get colder as it absorbs radiation. Because any radiation (or anything else) it absorbs increases its mass, and since higher mass BHs are colder, the more energy you dump into one, the colder it gets!