
\subsection{Expansion of BH thermodynamics} The original first law of BH thermodynamics, that exploits the quanutm relation between temperature \emph{T} and surface gravity  $\kappa$ and that of entropy \emph{S} and area of the black hole, for a charged rotating BH has the following form :
\begin{equation} \label{eqn : 1}
    \delta M = T\delta S + \Omega\delta J + \phi \delta Q \ ,
\end{equation} 
where \emph{M} is the mass identified with the internal energy \emph{U}, $\Omega$ and \emph{J} is angular velocity and angular momentum associated with rotating black hole and $\phi$ and \emph{Q} is the electric potential and charge associated with the charged BH. But equation \ref{eqn : 1} does not yet include the \emph{pressure} and \emph{volume} terms which are commonly found in thermodynamics. It was by then well established that the cosmological constant $\Lambda$ can be treated as a thermodynamical BH parameter \cite{16,17}. Consequently, the \emph{pressure} of the BH was interpreted in terms of a negative cosmological constant and the corresponding $P \delta V$ term was finally added to the original first law  \cite{18}, leading to  a more generalized first law of the following form :
\begin{equation}
    \delta M = T\delta S + V \delta P + \Omega\delta J + \phi \delta Q\ ,
\end{equation} where \begin{equation*}
 P = \frac{-\Lambda}{8\pi}
\end{equation*} and the corresponding conjugate variable volume is \cite{19,20},
\begin{equation*}
V \equiv \left( \frac{\partial M}{\partial P}\right)_{S,Q,J}
\end{equation*} This also changed the interpretation of mass \emph{M} to the enthalpy \emph{H} \cite{21} of the BH, which now involves both the internal energy \emph{U} and energy due to the \emph{PV} term. Expansion of BH thermodynamics in such a way leads to a new extended picture of the phase space that allows us to study phase transitions with a renewed perspective. Such an investigation has led to an enhanced identification between the AdS black holes and van der Waals liquid-gas system \cite{22,23} and other interesting discoveries \cite{24,25,26,27,28}.
\subsection{Holographic heat engines}
With this kind of framework at hand, Johnson put forth the idea of "holographic heat engines" \cite{29}, wherein he utilises the \emph{pdV} term to obtain mechanical work from heat energy, black holes being the working substance. For such a thermodynamic system, work can be defined in terms of a cycle in the \emph{P - V} plane with \emph{$Q_H$} and \emph{$Q_C$} being the net input and output heat respectively and the net mechanical work is written as \emph{W} =   \emph{$Q_H$} - \emph{$Q_C$}. The heat engine's efficiency is defined in the usual thermodynamical sense as $\eta = W/Q_H = 1 - Q_C / Q_H$. As we know, the efficiency of Carnot's engine corresponds with the maximum achievable efficiency given by,
\begin{equation*}
    \eta_c = 1 - \frac{Q_C}{Q_H} = 1 - \frac{T_C}{T_H}
\end{equation*} where $T_C$ and $T_H$ are lower and higher temperature of the reservoir respectively. \par In this paper, we develop a more generalised version of thermodynamic heat engine by incorporating the idea of massive gravity and study the effects of such a modification on the efficiency of the heat engine. There are several arguments one can make in support of such a generalisation : The theory of general relativity (GR) is known to describe the gravitational force with a good agreement with the observations. However the search for alternative theories has been an on-going quest since the formulation of GR. From a modern particle physics
perspective, GR is a unique theory for a massless spin-2 particle. Taking GR as starting point, alternative theories intend to modify or extend GR to different scales in a way as to account for the observational anomalies. One such modification maintains the notion of gravity being propagated by a spin-2 particle while giving a mass to this particle, that is the so called "Massive gravity". The first attempt to add a mass to the graviton was made by Fierz and Pauli \cite{30}, who in their work showed the presence of an extra degree of freedom (called ghost) in a generic Lorentz-invariant modification of gravity. Unfortunately, presence of an unwanted mode was discovered, named as the vDVZ discontinuity \cite{31,32} that represented failure of a massive gravity theory to reduce itself into general relativity (a massless spin-2 field theory with two propagating degree of freedom ) in the massless limit. This pathology naturally posed a threat to one of the most important underlying principle when it comes to deforming a theory : Continuity in physical predictions of a theory in the parameters of the theory. This issue was then resolved by Vainshtein in \cite{33} by proposing a mechanism through which nonlinearities managed to hid the extra degrees of freedom, at short distances and has been successfully applied to several models \cite{34,35,36}.
\par Thus, to summarize, the cosmological models motivated by GR are plagued by issues like the
cosmological constant problem, late time acceleration of universe, issues at the interface of gravity and quantum mechanics, etc., forcing us to look for modifications such as mentioned above. Massive gravity theories are of immense interest as an alternative to GR, strongly motivated on both theoretical and phenomenological grounds \cite{37,38,39}.
 Undoubtedly, its inclusion in the study of the efficiency of BH heat engines would result into some interesting insights. Also there are enough evidences of such impact of a massive graviton, for example the presence of van der Waals phase transitions for non-spherical geometries, in holography \cite{40}, in the resolution to the cosmological constant problem \cite{41,42}, for enhanced polarization of gravitational waves \cite{43} and many more. It would be interesting to find out the effect of such highly influential graviton mass on the efficacy of the heat engines. We consider a regular charged black hole (RCB) solution for such an investigationand and for additional work along this direction for different black hole systems see\cite{44,45,46}. In doing so, we also calculate temperature, pressure and specific heat of the RCB heat engine. \par The paper is organised as follows : In the next section, we begin by constructing a RCB heat engine in massive gravity in extended phase space and draw some important results through their plots. In section \ref{section :3} we turn to our main aim that is to study the effect of inclusion of graviton mass on the efficiency of the RCB heat engine. In the last section, we summarize our results.


\section{Introduction}
\input{Introduction}
\section{Regular charged black holes in massive gravity as heat engines}
\input{RCB}
\section{Efficiency of the RCB heat engine}
\input{effiiciency}
\newpage
\section{Conclusion}
\input{Conclusions}
