Thermodynamic formulation of Cotton gravity in the Codazzi parametrization

TL;DR

This paper formulates Cotton gravity thermodynamically using the Codazzi parametrization, applying it to cosmological and static spacetimes, and reveals how horizon thermodynamics can diagnose matter content.

Contribution

It introduces a thermodynamic framework for Cotton gravity in the Codazzi parametrization, linking horizon thermodynamics to matter diagnostics.

Findings

Derived a modified holographic entropy with Codazzi tensor corrections.

Identified the sign of the correction as a matter content diagnostic.

Applied the formalism to FRW and static spherically symmetric spacetimes.

Abstract

We develop a thermodynamic formulation of Cotton gravity in the Codazzi parametrization, providing a general framework in which the gravitational dynamics can be interpreted in terms of horizon thermodynamics. As paradigmatic examples, we apply the formalism to FriedmannRobertson-Walker (FRW) and static spherically symmetric spacetimes. By implementing the first law of thermodynamics on the apparent cosmological and event horizons, we derive a modified holographic entropy consisting of the standard Bekenstein-Hawking term supplemented by a correction induced by the Codazzi tensor. In the cosmological setting, this correction is governed by the temporal component of the Codazzi tensor, while in static configurations it is controlled by its anisotropic sector. Remarkably, the sign of this contribution provides a potential diagnostic of the underlying matter content, allowing one to distinguish between ordinary matter, a cosmological constant and phantom-like components. These results establish horizon thermodynamics as a sensitive probe of Cotton gravity, offering a complementary perspective beyond background kinematics and enabling a characterization of the statistical and thermodynamic properties of spacetime within the Codazzi formulation.