Thermodynamic analysis of a compact object in Rastall-Rainbow gravity

TL;DR

This study explores the thermodynamics of horizonless compact objects in Rastall-Rainbow gravity, revealing stable remnants, phase branches, and the distinct roles of Rastall and Rainbow parameters in thermodynamic stability.

Contribution

It provides the first detailed thermodynamic analysis of gravastars in Rastall-Rainbow gravity, showing how modifications lead to stable remnants and influence phase stability.

Findings

RR modifications produce two temperature extrema and a stable zero-temperature remnant.

Small RR gravastars are thermodynamically favored over hot curved space.

Increasing Rastall and Rainbow parameters differently affects stability and temperature behavior.

Abstract

In this paper, we investigate the thermodynamic behavior of a horizonless compact object within the framework of Rastall-Rainbow (RR) gravity. Working with local shell thermodynamics for gravastar and an exterior fiducial temperature, we show that the RR modification bends temperature to produce two extrema and a stable mass remnant at zero temperature. We show that the gravastar's shell entropy is smaller than that of a comparable black hole, and that RR modifications introduce a logarithmic correction which contributes to specific heat positivity and a smoother free energy landscape of small gravastars. A Central finding of this work is that, from heat capacity and Helmholtz free energy analyses, we uncover small, middle, and large branches and demonstrate that unlike Rainbow modified black holes, the small RR gravastar is both locally and globally favored over hot curved space. At the parameter level, both Rastall and Rainbow play distinct roles. Increasing the Rastall parameter, by strengthening matter-curvature coupling, adjusts the redshift between the shell and the exterior, shifts the temperature maximum to higher values at larger masses, and narrows the unstable window. In contrast, increasing the Rainbow parameter enhances energy dependent UV suppression and bends the temperature in lower values at larger masses. Altogether, these results highlight a controlled route to thermodynamic stabilization and the emergence of a stable remnant in horizonless compact objects within RR gravity.