  A spatial variant of the Functional Renormalization Group (FRG) is introduced
on (Lorentzian signature) globally hyperbolic spacetimes. Through its
perturbative expansion it is argued that such a FRG must inevitably be state
dependent and that it should be based on a Hadamard state. A concrete
implementation is presented for scalar quantum fields on flat
Friedmann-Lema\^{i}tre spacetimes. The universal ultraviolet behavior of
Hadamard states allows the flow to be matched to the one-loop renormalized flow
(where strict removal of the ultraviolet cutoff requires a tower of potentials,
one for each power of the Ricci scalar). The state-dependent infrared behavior
of the flow is investigated for States of Low Energy, which are Hadamard states
deemed to be viable vacua for a pre-inflationary period. A simple
time-dependent infrared fixed point equation (resembling that in Minkowski
space) arises for any scale factor, with analytically computable corrections
coding the non-perturbative ramifications of the Hadamard property in the
infrared.
