Thermodynamic analysis of shift-symmetric black-hole spacetimes in Horndeski gravity

In this study, we investigate the thermodynamic properties of shift-symmetric Horndeski and beyond-Horndeski theories (theories in which only derivatives of the scalar field appear in the action). Utilizing Euclidean methods, we first analyze two specific cases that serve as foundational examples for the broader framework. We derive the general expression for entropy variation within this setting, for homogeneous spacetimes (with ${g}_{tt}=1/{g}_{rr}$), demonstrating that both parity-preserving and parity-violating terms contribute to the entropy formula. Given the functional form of the coupling terms, the black-hole entropy can be determined by integrating this relation with respect to the event horizon. Our general result is consistent with previously reported special cases in the literature. Notably, for particular constraints on the coupling functions, the entropy identically vanishes. Furthermore, we establish that homogeneous spacetimes within shift- and parity-symmetric beyond-Horndeski theories universally obey Bekenstein's area law for entropy, irrespective of the specific form of the coupling functions.