Is the rate of acceleration of expansion of the universe known?
I think I may be able to provide some detail, but am no expert.
The rate of expansion of the universe is tricky as there isn’t a physical border (sudden, definite change in density) that a signal can reflect off. We infer it by measuring the speed of recession of the most distant galaxy clusters and the (even more visible) quasars and gamma ray bursts.
These are known because we can relate known electronic transitions (an electron in an atom produces light of a single frequency, enabling us to identify it when it moves from one orbit to a lower one) to their recession velocity by the Doppler effect (See here https://en.m.wikipedia.org/wiki/Doppler_effect)
Hope this at least explains the principles.