  We analyze the approach to chiral symmetry breaking in QCD at finite
temperature, using the functional renormalization group. We compute the running
gauge coupling in QCD for all temperatures and scales within a simple truncated
renormalization flow. At finite temperature, the coupling is governed by a
fixed point of the 3-dimensional theory for scales smaller than the
corresponding temperature. Chiral symmetry breaking is approached if the
running coupling drives the quark sector to criticality. We quantitatively
determine the phase boundary in the plane of temperature and number of flavors
and find good agreement with lattice results. As a generic and testable
prediction, we observe that our underlying IR fixed-point scenario leaves its
imprint in the shape of the phase boundary near the critical flavor number:
here, the scaling of the critical temperature is determined by the
zero-temperature IR critical exponent of the running coupling.
