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.

中文翻译：

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大重力下的临界热机

在扩展热力学中，我们研究了球形（$k=1$）、平面（$k=0$）和双曲线（$k=- 1$) 拓扑。尽管对于双曲（球形）拓扑，$\eta_k$ 通常较高（较低），但我们表明，该顺序可以在临界热机中颠倒，球形拓扑的效率更高，特别是以下顺序：$ \eta_{\ rm -1}^{\phantom{-1}} < \eta_{\rm 0}^{\phantom{0}} < \eta_{\rm +1}^{\phantom{+1}}$。此外，对临界孔近地平线区域的研究表明，除了已知的 $q\rightarrow\infty $ 条件外，还需要根据几何拓扑对大量重力参数进行额外缩放，