Thermodynamics and $P$-$V$ Criticality of Charged AdS Black Holes with a Cloud of Strings in Kalb-Ramond Gravity

TL;DR

We analyze electrically charged AdS black holes in Einstein--Kalb--Ramond (EKR) gravity with a spherically symmetric cloud of strings in the extended phase space. The lapse function contains Lorentz-violating parameter $\ell$ and string-cloud parameter $\alpha$, leading to a thermodynamic structure where the ADM mass is enthalpy and $P=-\Lambda/(8\pi(1-\ell))$ acts as pressure; the equation of state exhibits Van der Waals–like behavior with a universal ratio $\frac{P_c v_c}{T_c}=\tfrac{3}{8}$, while critical scales $v_c,T_c,P_c$ depend on $\ell$ and $\alpha$. Thermodynamic topology yields a total charge $W=1$, placing these black holes in the same topological class as RN--AdS, with LV and string-cloud effects shifting zero points but not the class. The Joule–Thomson analysis shows the minimal inversion temperature satisfies $T_i^{\min}/T_c=\tfrac{1}{2}$, identical to RN--AdS, indicating that universality in JT behavior is preserved despite the new parameters. Overall, a flexible framework emerges where Lorentz violation and string-cloud matter deform thermodynamic scales without altering the underlying universality class or topology.

Abstract

We investigate the extended phase-space thermodynamics and $P$--$V$ criticality of electrically charged anti-de Sitter (AdS) black holes in Kalb--Ramond bumblebee gravity in the presence of a spherically symmetric cloud of strings. The background Kalb--Ramond field induces Lorentz symmetry violation through a dimensionless parameter $\ell$, while the string cloud is characterized by a parameter $α$, both entering the lapse function and deforming the geometry. Interpreting the ADM mass as enthalpy, we derive the main thermodynamic quantities, Hawking temperature, entropy, thermodynamic volume, Gibbs free energy, internal energy, and specific heat, and analyze how $(\ell,α)$ jointly affect stability and phase structure. We obtain a Van der Waals--type equation of state, compute the critical point $(P_c,T_c,v_c)$, and show that, although the critical scales depend nontrivially on $\ell$ and $α$, the universal ratio $P_c v_c/T_c = 3/8$ is preserved. Using the thermodynamic topology approach, we determine that the total topological charge of the black hole solution remains $W=1$, placing it in the same topological class as the Reissner--Nordström--AdS black hole. Finally, by studying the Joule--Thomson expansion, we derive the inversion curve and find that the minimal inversion temperature satisfies $T_i^{\rm min}/T_c = 1/2$, as in the Reissner--Nordström--AdS case, indicating that Lorentz violation and the string cloud deform the thermodynamic scales without changing the underlying universality class.

Figures (10)

• Figure 1: ADM mass $M$ as a function of the horizon radius $r_+$ for a charged AdS black hole in EKR bumblebee gravity with a cloud of strings. Panel (i): fixed $\ell=-0.1$ and several values of the string cloud parameter $\alpha$. Panel (ii): fixed $\alpha=0.1$ and several values of the LV parameter $\ell$. In both panels we set $Q=0.5$ and $\Lambda=-0.003$.
• Figure 2: Hawking temperature $T_H$ as a function of the horizon radius $r_+$ for a charged AdS black hole in EKR bumblebee gravity with a cloud of strings. Panel (i): fixed $\ell=-0.1$ and different values of the string cloud parameter $\alpha$. Panel (ii): fixed $\alpha=0.1$ and different values of the LV parameter $\ell$. In both cases we set $Q=0.5$ and $\Lambda=-0.003$. The deformation of the $T_H(r_+)$ curves relative to the Reissner--Nordström--AdS case illustrates the impact of $\ell$ and $\alpha$ on the small- and large-black-hole branches.
• Figure 3: Gibbs free energy $F$ as a function of the horizon radius $r_{+}$ for a charged AdS black hole in EKR bumblebee gravity with a cloud of strings. Panel (i): fixed $\ell=-0.1$ and different values of $\alpha$. Panel (ii): fixed $\alpha=0.1$ and different values of $\ell$. Here $Q=0.5$ and $\Lambda=-0.003$. The shifts in $F(r_+)$ show how the LV and string parameters influence the relative thermodynamic preference of different horizon sizes.
• Figure 4: Internal energy $U$ as a function of the horizon radius $r_{+}$ for $Q=0.5$. Panel (i): fixed $\ell=-0.1$ and varying $\alpha$. Panel (ii): fixed $\alpha=0.1$ and varying $\ell$. The presence of the string cloud and Lorentz violation modifies both the slope and magnitude of $U(r_+)$, effectively "dressing" the black hole energy for a given horizon size.
• Figure 5: Specific heat capacity $C_{\rm heat}$ at constant pressure as a function of the horizon radius $r_{+}$, for $\Lambda=-0.003$ and $Q=3$. Panel (i): fixed $\ell=-0.1$ and varying $\alpha$. Panel (ii): fixed $\alpha=0.1$ and varying $\ell$. The zeros and divergences of $C_{\rm heat}$ delimit stable ($C_{\rm heat}>0$) and unstable ($C_{\rm heat}<0$) black hole branches, and their dependence on $\ell$ and $\alpha$ encodes the influence of Lorentz violation and the string cloud on local thermodynamic stability.