  A principled framework to generalize variational perturbation approximations
(VPA's) formulated within the ambit of the nonadditive statistics of Tsallis
statistics, is introduced. This is accomplished by operating on the terms
constituting the perturbation expansion of the generalized free energy (GFE)
with a variational procedure formulated using \emph{q-deformed calculus}. A
candidate \textit{q-deformed} generalized VPA (GVPA) is derived with the aid of
the Hellmann-Feynman theorem. The generalized Bogoliubov inequality for the
approximate GFE are derived for the case of canonical probability densities
that maximize the Tsallis entropy. Numerical examples demonstrating the
application of the \textit{q-deformed} GVPA are presented. The qualitative
distinctions between the \textit{q-deformed} GVPA model \textit{vis-\'{a}-vis}
prior GVPA models are highlighted.
