  We use a combination of first principles many-body methods and the numerical
renormalization-group technique to study the Kondo regime of cobalt-porphyrin
compounds adsorbed on a Cu(111) surface. We find the Kondo temperature to be
highly sensitive to both molecule charging and distance to the surface, which
can explain the variations observed in recent scanning tunneling spectroscopy
measurements. We discuss the importance of many-body effects in the molecular
electronic structure controlling this phenomenon and suggest scenarios where
enhanced temperatures can be achieved in experiments.
