Is Ch. 6 of Birrell & Davies book on QFT in curved space and in particular the 1-loop effective action that they derive up-to-date with the current state of the art in (effective) quantum gravity?
I have no background in string theory and only in condensed-matter and I'm quite liking the approach from the book. Is there something important that I should be aware that appeared since the book was published?
Objectively speaking, the best book on QFT in curved spacetime is DeWitt's The Global Approach to Quantum Field Theory (2003). For one thing, it was written by one of the founding fathers of the subject. In this book you will find the most general and systematic formulation of an arbitrary Quantum Field Theory.
The author uses functional methods from the outset, so everything is explicitly covariant. Furthermore, the spacetime manifold is left arbitrary (both its geometry and its topology). Similarly, the fields and their dynamics are also arbitrary: they can be either fermionic or bosonic, have any spin, and be gauge fields (corresponding to an arbitrary algebra, not necessarily closed or irreducible). In this sense, the formulation is as general as possible.
In the book you will find a discussion of essentially every topic of QFT and, in particular, of quantum theories in a curved background (dynamical vacuum and its thermal properties, black-holes, etc.). You will also find a (somewhat idiosyncratic but still very informative) discussion of the quantisation of the gravitational field itself.
The mathematics are very rigorous (up to physicists standards) and precise. Unfortunately, the non-trivial geometry of the manifold seems to preclude a straightforward implementation of the programme introduced by Epstein and Glaser, so one cannot proceed by a completely rigorous formulation. Therefore, the author anticipates (and finds) UV divergences, as is usual in introductory textbooks. Nevertheless, the analysis of divergences is as general as possible, so that the formulation is rather convincing anyway. If you want generality and completeness, you really can't do better than this book. A must-read indeed!
For more mathematically oriented readers, I cannot help but recommend R. Brunetti, C. Dappiaggi, K. Fredenhagen & J. Yngvason's Advances in Algebraic Quantum Field Theory (2015) (with the collaboration of our very own V. Moretti!). In this book you will find a very thorough and up-to-date discussion of AQFT and its applications to, among others, quantum field theory in a curved background. Along the same lines, and as mentioned in the comments, Wald has dedicated several papers to the matter, so make sure to check them out.
Finally, the Wikipedia page on QFT in curved spacetime contains a list of many good references that you should check out too. Good luck!
