  With in the extended thermodynamics, we study the efficiency $\eta_k$ of
critical heat engines for charged black holes in massive gravity for spherical
($k=1$), flat ($k=0$) and hyperbolic ($k=-1$) topologies. Although, $\eta_k$ is
in general higher (lower) for hyperbolic (spherical) topology, we show that
this order can be reversed in critical heat engines with efficiency higher for
spherical topology, following in particular the order: $ \eta_{\rm
-1}^{\phantom{-1}} < \eta_{\rm 0}^{\phantom{0}} < \eta_{\rm
+1}^{\phantom{+1}}$. Furthermore, the study of the near horizon region of the
critical hole shows that, apart from the known $q\rightarrow \infty $
condition, additional scalings of massive gravity parameters, based on the
topology of the geometry are required, to reveal the presence of a fully
decoupled Rindler space-time with vanishing cosmological constant.
