Black dwarfs are thought to be the end stage of stars that initially have between .07 and 10 solar masses. After passing through a red giant phase, a significant proportion of the mass of the star is shed as a planetary nebula. What remains is a white dwarf, which is initially extremely hot, but which does not generate any new energy through nuclear fusion -- and which is supported by electron degeneracy pressure. White dwarfs cool very slowly, eventually becoming dead black dwarfs. This process takes so long that the universe is not thought to be old enough to contain black dwarfs yet, so they are theoretical objects.
A black dwarf is a white dwarf that has cooled down to the temperature of the cosmic microwave background, and so is invisible. Unlike red dwarfs, brown dwarfs, and white dwarfs, black dwarfs are entirely hypothetical.
Once a star has evolved to become a white dwarf, it no longer has an internal source of heat, and is shining only because it is still hot. Like something taken from the oven, left alone a white dwarf will cool down until it is the same temperature as its surroundings. Unlike tonight’s dinner, which cools by convection, conduction, and radiation, a white dwarf cools only by radiation.
Because it’s electron degeneracy pressure that stops it from collapsing to become a black hole, a white dwarf is a fantastic conductor of heat . How fast a white dwarf cools is thus easy to work out … it depends on only its initial temperature, mass, and composition (most are carbon plus oxygen; some maybe predominantly oxygen, neon and magnesium; others helium). At least, part of the core of a white dwarf may crystallize, the cooling curve will have a bit of a bump around then.
The universe is only 13.7 billion years old, so even a white dwarf formed 13 billion years ago (unlikely; the stars which become white dwarfs take a billion years, or so, to do so) it would still have a temperature of a few thousand degrees. The coolest white dwarf observed to date has a temperature of a little less than 3,000 K. A long way to go before it becomes a black dwarf.