  We study the behavior of the entropy of the pseudogap Bose-Fermi Kondo model
within a dynamical large-$N$ limit, where $N$ is related to the symmetry group
of the model. This model is a general quantum impurity model that describes a
localized level coupled to a fermionic bath having a density of states that
vanishes in a powerlaw fashion near the Fermi energy and to a bosonic bath
possessing a powerlaw spectral density below a cutoff energy. As a function of
the couplings to the baths various quantum phase transitions can occur. We
study how the impurity entropy changes across these zero-temperature
transitions and compare our results with predictions based on the g-theorem.
This is accomplished by an analysis of the leading and sub-leading scaling
behavior. Our analysis shows that the $g$-theorem does not apply to the
pseudogap Bose-Fermi Kondo model at the large-N level. This inapplicability
originates from an anomalous contribution to the scaling function in the
hydrodynamic regime where $k_B T>\hbar \omega$ which is absent in the quantum
coherent regime, i.e., for $k_B T<\hbar \omega$. We also compare our results
with those obtained for the Sachdev-Ye-Kitaev model.
