  A toy model of Einstein gravity with a Gauss-Bonnet classically "entropic"
term mimicking a quantum correction is considered. The static black hole
solution due to Tomozawa is found and generalized with the inclusion of non
trivial horizon topology, and its entropy evaluated deriving the first law by
equations of motion. As a result the Bekenstein-Hawking area law turns to be
corrected by a logarithmic area term. A Misner-Sharp expression for the mass of
black hole is found. Within a Friedmann-Lema\^itre-Robertson-Walker (FLRW)
cosmological setting, the model is used in order to derive modified Friedmann
equations. Such new equations are shown to reproduce the first law with the
same formal entropy and quasi local energy of the static case, but here within
a FLRW space-time interpreted as a dynamical cosmological black hole. A
detailed analysis of cosmological solutions is presented, and it is shown that
the presence of the correction term provides regular solutions and interesting
phases of acceleration and decelerations, as well as, with negligible matter,
exact de Sitter solutions.
