I couldn't right off hand find anything about the future of the CMBR temperature. But I did find this about the future of the universe. WIKI:Future of an expanding universe

Originally Posted By: Wikipedia
Black Hole Era

10^40 years to 10^100 years

After 10^40 years, black holes will dominate the universe. They will slowly evaporate via Hawking radiation. A black hole with a mass of around 1 solar mass will vanish in around 2×10^66 years. As the lifetime of a black hole is proportional to the cube of its mass, more massive black holes take longer to decay. A supermassive black hole with a mass of 10^11 (100 billion) solar masses will evaporate in around 2×10^99 years.


I'm not sure whether the CMBR is counted in that, but it does give an estimate of the future heat death of the universe.

Bill Gill

C is not the speed of light in a vacuum.
C is the universal speed limit.