Consider the following paragraph taken from page 30 of Thomas Hartman's lecture notes on Quantum Gravity:
Hawking radiation is a feature of QFT in curved spacetime. It does not require that we quantize gravity - it just requires that we quantize the perturbative fields on the black hole background. In fact we can see very similar physics in at spacetime.
Does a QFT in curved spacetime simply replace a Lagrangian in flat Minkowski spacetime with the same Lagrangian in a given curved spacetime?
Does a QFT in curved spacetime not include the Einstein action?
Does a QFT in curved spacetime have limited predictive power than the non-renormalizable quantized Einstein-Hibert action coupled to the same QFT, because we have to choose a specific spacetime metric in the former case in order to draw predictions from the theory?
Does a QFT in curved spacetime quantize the matter fields which manifest themselves well below the Planck scale, but use a given classical spacetime metric?