Why are black dwarves so cold?

I’m going to reveal a gap in my understanding of stellar evolution (or, more likely, physics) that I’m hoping someone here can fill. Without fusion to heat it, a white dwarf radiates its latent heat into space, cooling over hundreds of billions of years, to become a black dwarf. Ultimately, the black dwarf may become as cold as 5 degrees K. On the other hand, the pressure in the core of the black dwarf is enormous due to its immense gravity (the result of, say, half the mass of the Sun packed into the volume of the Earth). Doesn’t this pressure create enormous heat? If so, where does it go? If not, why not?

In other other words, if gravitationally-induced pressure can raise the temperature of the core of a gas cloud high enough to trigger fusion, why doesn’t it keep the core of a black dwarf warm indefinitely, which would seem to violate the first law of thermodynamics?