Which stage follows the white dwarf stage of star development?
Theoretically the Black Dwarf stage.
A white dwarf star does not undergo fusion anymore, therefore it no longer generates energy. It does however, still have an awful lot of heat to it that slowly bleeds off into space. The oldest, and therefore coolest, white dwarf star known to man still has a surface temperature over 3000 degrees K.
Once a white dwarf has cooled off to the point that it is the same temperature as background space (3 K roughly), it is no longer radiating heat of any sort and at that point is considered a black dwarf.
The reason that I said it was theoretically, is that the Universe itself is not old enough for a star to have formed, died, and subsequently radiated away all it's heat. The universe is not even 14 billion years old and the estimated shortest time frame to create a black dwarf is one quadrillion years, or about 70,000 times longer than the universe has been in existence. If our species were ever to see a black dwarf it would be long after our sun had gone nova and turned into a white dwarf.