  We investigate the phenomenon of gravitational catalysis, i.e.,
curvature-induced chiral symmetry breaking and fermion mass generation, at
finite temperature. Using a scale-dependent analysis, we derive a thermal bound
on the curvature of local patches of spacetime. This bound quantifies regions
in parameter space that remain unaffected by gravitational catalysis and thus
are compatible with the existence of light fermions as observed in Nature.
While finite temperature generically relaxes the curvature bound, we observe a
comparatively strong dependence of the phenomenon on the details of the
curvature.
  Our bound can be applied to scenarios of quantum gravity, as any realistic
candidate has to accommodate a sufficient number of light fermions. We argue
that our bound therefore represents a test for quantum gravity scenarios: a
suitably averaged spacetime in the (trans-)Planckian regime that satisfies our
curvature bound does not induce correspondingly large Planckian fermion masses
by gravitational catalysis. The temperature dependence derived in this work
facilitates to follow the fate of gravitational catalysis during the thermal
history of the (quantum) universe. In an application to the Asymptotic Safety
scenario of quantum gravity, our bound translates into a temperature-dependent
upper bound on the number of fermion flavors.
