As the title suggests. I think it should, as changing flat spacetime to slightly non-flat spacetime should not make our theory invalid, as our world is only approximatively flat, and this should have been already taken care of. But I am not sure in case where gravity matters strongly, where new divergences become apparent, and then it may be possible that infinitely many physical parameters are needed to transform our action to be physical, which would mean a theory is not renormalizable.
So restricting to defining QFT on globally hyperbolic curved spacetime, what happens for renormalization? If everything is renormalizable up to reasonable standards we have for QFT in flat spacetime - after all, we do not have a rigorous proof that renormalization works up to arbitrary high energy even in flat spacetime - why is this the case?
