Thermodynamic Analysis of Charged AdS Black Holes with Cloud of Strings in Einstein-Bumblebee Gravity via Tsallis Entropy

Abstract

We investigate the thermodynamic properties of charged anti-de Sitter black holes surrounded by a cloud of strings in bumblebee gravity. In this framework, the cloud-of-strings parameter $α$ and the Lorentz-violating parameter $\ell$ modify the horizon structure, the Hawking temperature, the free energies, the specific heat, and the critical behavior in the extended phase-space description. We derive the corresponding equation of state and show that the system exhibits a small--large black-hole phase transition of Van der Waals type. In particular, the critical quantities are deformed by both the cloud of strings and the bumblebee background, while the universal ratio is explicitly altered by Lorentz symmetry breaking. We also examine the Joule--Thomson expansion and analyze the associated inversion and isenthalpic curves, showing how the deformation parameters shift the boundary between heating and cooling regions. In addition, we extend the thermodynamic analysis to a Tsallis entropy-based framework and show that the non-extensive parameter $δ$ significantly changes the temperature profile, stability windows, critical volume, free energies, and sparsity of Hawking radiation. Our results reveal that the combined effects of the string cloud, Lorentz violation, and non-extensive entropy lead to a substantially richer thermodynamic structure than that of the standard Reissner--Nordström--AdS black hole.

Figures (13)

• Figure 1: Hawking temperature $T_H$ as a function of the horizon radius $r_h$. Panel (a) shows the effect of the Lorentz-violating parameter $\ell$ for fixed $\alpha=0.2$, $q=1$, and $P=0.015$. Panel (b) shows the effect of the cloud-of-strings parameter $\alpha$ for fixed $\ell=0.3$, $q=1$, and $P=0.015$. The curves show that increasing $\ell$ suppresses the temperature through the global factor $(1+\ell)^{-1/2}$, while increasing $\alpha$ lowers the overall temperature scale by reducing the effective contribution of the $(1-\alpha)$ term.
• Figure 2: Specific heat at constant pressure $C_P$ as a function of $r_h$. Panel (a) shows the dependence on the Lorentz-violating parameter $\ell$ for fixed $\alpha=0.2$, $q=1$, and $P=0.015$, while panel (b) shows the dependence on $\alpha$ for fixed $\ell=0.3$, $q=1$, and $P=0.015$. The vertical divergences correspond to Davies points, where the denominator of Eq. (\ref{['heat-1']}) vanishes and the black hole changes from a locally unstable branch ($C_P<0$) to a locally stable branch ($C_P>0$).
• Figure 3: Gibbs free energy $G$ as a function of Hawking temperature $T_H$. Panel (a) shows the effect of $\ell$ for fixed $\alpha=0.2$, $q=1$, and $P=0.012$, whereas panel (b) shows the effect of $\alpha$ for fixed $\ell=0.3$, $q=1$, and $P=0.012$. The swallowtail-like behavior signals the presence of competing thermodynamic branches and is the standard signature of a first-order small--large black-hole phase transition in the extended phase-space picture.
• Figure 4: $P$--$v$ isotherms for four representative choices of the deformation parameters: (a) $\ell=0$, $\alpha=0$; (b) $\ell=0.5$, $\alpha=0$; (c) $\ell=0$, $\alpha=0.3$; and (d) $\ell=0.5$, $\alpha=0.3$, with $q=1$ in all panels. Each panel includes isotherms below, at, and above the critical temperature, together with the critical point marked by a star. The subcritical curves show the Van der Waals-like oscillatory behavior associated with a small--large black-hole phase transition.
• Figure 5: Critical quantities in the standard thermodynamic framework. Panel (a) shows the universal critical ratio $\rho_c=P_c v_c/T_c=3\sqrt{1+\ell}/8$ as a function of the Lorentz-violating parameter $\ell$ for $q=1$. Since $\rho_c$ is independent of the cloud-of-strings parameter $\alpha$, all curves corresponding to different $\alpha$ coincide exactly; for this reason, only the single resulting profile is displayed. The horizontal dotted line indicates the standard RN--AdS value $\rho_c=3/8$. Panel (b) shows the normalized critical quantities $v_c/v_c^{(0)}$, $T_c/T_c^{(0)}$, and $P_c/P_c^{(0)}$ as functions of $\ell$ for fixed $\alpha=0.2$ and $q=1$.