Richard Woodard and his collaborators (notably N. Tsamis) have performed many non-trivial calculations relevant for the QFT of GR.
Probably the most intriguing result is that quantum effects lead to an end of inflation. See also this version. The calculation is very difficult and I do not know if the interpretation is correct or not. But this deserves to be understood better.
Here is a paper by Jaume Garriga and T. Tanaka which criticizes the Tsamis Woodard screening mechanism.