  We discuss the scaling of the effective action for the interacting scalar
quantum field theory on generic spacetimes with Lorentzian signature and in a
generic state (including vacuum and thermal states, if they exist). This is
done constructing a flow equation, which is very close to the renown Wetterich
equation, by means of techniques recently developed in the realm of
perturbative Algebraic Quantum Field theory (pAQFT). The key ingredient that
allows one to obtain an equation which is meaningful on generic Lorentzian
backgrounds is the use of a local regulator, which keeps the theory covariant.
As a proof of concept, the developed methods are used to show that non-trivial
fixed points arise in quantum field theories in a thermal state and in the case
of quantum fields in the Bunch-Davies state on the de Sitter spacetime.
