You are correct: a black hole whose Hawking temperature is less than that of the cosmic background radiation is “colder” than the space it is immersed in and co

You are correct: a black hole whose Hawking temperature is less than that of the cosmic background radiation is “colder” than the space it is immersed in and consequently absorbs energy (and mass-energy equivalence) from the heat bath it is immersed in. The easiest way to envision it is that the black hole swallows up photons of the microwave background radiation and is emitting less energy as Hawking radiation, leading to a net imbalance in the black hole’s favour. Of course eventually the expanding universe will drop the temperature of the cosmic background radiation further and the process will reverse.