  We study the evolution of a metric of a two dimensional black hole under the
second loop renormalization group fow, the RG-2 fow. Since the black hole
metric is noncompact (we consider it asymptotically flat) we adapt some proofs
for the compact case to the asymptotically flat case. We found that the
appearance of horizons during the evolution is related to the condition of
parabolicity of the flow. We also show that the entanglement entropy of a two
dimensional black hole is monotonic under the RG-2 flow. We generalize the
results obtained for the first loop approximation and discuss the implications
for higher order loops.
