  The functional renormalization group treatment of the conform reduced
Einstein-Hilbert gravity is extended by following the evolution of the time and
space derivatives separately, in order to consider the Lorentz symmetry during
the evolution. We found the Reuter fixed point in the ultraviolet region. It is
shown that starting from the Gaussian fixed point the Lorentz symmetry breaks
down in the vicinity of the Reuter fixed point. Similarly, in the symmetry
broken phase it also breaks down in the infrared region close to a critical
singularity scale. By calculating the anomalous dimension form the kinetic term
of the action, we found a new relevant coupling belonging to the curvature.
