White dwarfs have good thermal conductivity, so the core won’t be much hotter than the surface.
Young white dwarfs start life at ~10 million K. But that phase doesn’t last long. Heat dissipation scales as T^4 (Stefan’s Law), so the initial cooling is very rapid.
A rough calculation suggests that if a white dwarf takes 10 billion years to cool to 10,000 K, the same white dwarf would take only about a decade to cool from 10^7 to 10^6 K, and a few hours to cool from 10^8 to 10^7 K.