How do astronomer determine the temperature of a black dwarf star?
They can't. A black dwarf is a star that takes trillions of years to develop and the Universe is currently 13.8 billion years old, so black dwarfs are a big mystery.
To give a brief idea, astronomers have theorized that white dwarfs, surviving for trillions of years, have exhausted their supplies, they eventually cool down and lose their energy. Over time, the white dwarf turns darker and darker, and its temperature drops.