The free energy of TAP-solutions for the SK-model of mean-field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical Sanov-type large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model. From a non-spin glass point of view, this work is the first in a series of refinements that addresses the stability of hierarchical structures in models of evolving populations. Based on joint work with N. Kistler and M.A. Schmidt.