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Basically if I shined a flashlight at a black hole, would I cause it's Hawking radiation temperature increase by more than the temperature of the light I shine at it, at any time during the life of that black hole? Or when CMB photons fall into the black hole are they returned to the universe hotter than the CMB?
Of course you see where I'm going with this. Is there a possibility that black holes counter the heat death?
Adding mass/energy to a black hole doesn't raise but (temporarily) lowers its Hawking temperature.
There's not really a sense of tracking individual photons that fall into a black hole and saying when "those" photons are reemitted. If a black hole starts out colder than the CMB (which all stellar-mass and larger black holes do), then it will cool further as it absorbs CMB radiation. The Hawking radiation that it emits at this stage will be cooler than the CMB. After a really long time, the CMB will be redshifted down to an effective temperature below the black hole's, and only then will the net flow of radiation start exiting the hole and it begins shrinking and heating up. The Hawking radiation that it emits at this stage will be hotter than the CMB.
I don't see how any of this could counter the heat death.
