By the time our Sun becomes a black dwarf, stellar evolution will be over. No new stars will form. Instead, the universe will fill up with cold remnants of the stars. That will allow the universe to start developing some odd stars that are quite different from what we know.
One is the frozen star. As the stars of the universe burn through their nuclear fuel, they will increase their metallicity. In astronomy, this is the measure of elements in a star that are heavier than helium—basically all the elements from lithium on. As the metallicity of the stars increase, they will get colder because heavier elements give off less energy from fusion. Eventually, stars will get so cold that they will have a temperature of 273 Kelvin, the freezing point for water.
Jumping ahead to the far distant future, an even weirder star will emerge. Approximately 101500 years in the future, entropy will have had its way and the universe will be essentially dead. In this cold time, quantum effects will rule the universe.
Quantum tunneling will then begin to allow light elements to fuse into an unstable form of iron. This will then decay into a more stable isotope, giving off faint amounts of energy. These so-called iron stars will be the only form of star possible in that time. However, they only occur in models where astronomers do not believe that protons will decay, so they are not a mainstream idea.