Q:

When will the last white dwarfs become black dwarfs?

I want my planet to be the last planet in a universe to host life forms. I expect a rough estimation as answer.
Thanks.

A:

About 100 trillion years from now.
The answer to your question depends on several things:

When the final low-mass star forms
How long it takes that star to exit the main sequence and become a white dwarf
The time needed for the white dwarf to cool to become a black dwarf

Given a typical galaxy and extrapolating from the current star formation rate, the final round of star formation should occur around 10-100 trillion years in the future, conservatively. Beyond this, degenerate remnants and brown dwarfs may collide from time to time, briefly producing an object that will heat up, but the main sequence lifetimes of the products will be in the range of tens of trillions of years, on the same order as our upper limit for normal star formation.
We now have to figure out how long it will take these final stars to become white dwarfs. Cooling models of the least massive of these stars indicate that a $\sim0.1M_{\odot}$ late M-type star will join the main sequence after two billion years, leave it roughly 6 trillion years later, and take about 500 billion years to cool to temperatures of about 1600 K, at which point it will be emitting very, very little visible light, and would likely be regarded as a black dwarf. The entire journey takes 6-7 trillion years.
Such a star won't be the greatest choice for supporting life. The M star in question should have a peak luminosity of $\sim0.01L_{\odot}$, requiring a very tight orbit for any potentially habitable exoplanet. During its white dwarf phase, it will plunge as low as $10^{-5}L_{\odot}$, pretty firmly taking it out of the habitability debate. Your planet may need extra energy sources throughout all parts of the star's life, and we have a few questions on that.
In short, the last black dwarf should form trillions of years after star formation ceases, which itself should happen 10-100 trillion years in the future.

