Now let's get one thing straight from the outset. Black dwarf stars don't exist and are completely theoretical. They are thought to be what the end product will be when a white dwarf has cooled sufficiently to have a temperature similar to that of the Cosmic Background Radiation. In this state, they would be essentially giving off no heat at all and thus would be pretty much invisible.
The reason that they don't exist is that white dwarves take an extraordinary long time to cool down - estimated at longer than the current age of the Universe. There's just not been enough time for any to form yet.
Certain parameters come into play when working out when a black dwarf star may result from a white one, many of which are unknown or speculative. We'll explore them here.
As we've established that white dwarves will have to have cooled to a temperature comparable to that of the Cosmic Background Radiation to become black dwarves and this is likely to only happen maybe after trillions of years, we have to consider how much further the CMB itself will have cooled by then.
As the Universe continues to expand, the temperature of the CMB decreases, albeit very slowly. That notwithstanding, it still means that a white dwarf has to cool even more before it reaches black dwarf status.
If it does indeed turn out that proton decay is a reality, this process within a white dwarf would serve to increase its temperature, thus slowing down the cooling process, as a result, lengthening its lifespan even further.
Some scientific reports researching dark matter have suggested that white dwarves may interact with, or contain this elusive mysterious substance. If this is the case, then the result would have the effect of keeping the white dwarf warmed up - thus hindering its metamorphosis into a black dwarf.
On their travels, white dwarves will inevitably interact with other celestial objects - other white dwarves, interstellar dust etc. Once again, this results in heating them up.
A white dwarf allowed to cool without any of the above factors coming into play will reach black dwarf status in anything up to several trillion years. The constant interactions outlined on this page will, as I've said, hinder this process, so a black dwarf star would only have any chance of existing some 1,000,000,000,000,000 years in the future. At this point, it will have cooled to about -270° Celsius.