# Friedmann Equations and Thermodynamics of Apparent Horizons

## TL;DR

This paper demonstrates that the first law of thermodynamics at the apparent horizon can be derived from the Friedmann equation across various gravity theories, revealing a deep physical connection.

## Contribution

It establishes a universal link between the Friedmann equations and thermodynamics of apparent horizons in multiple gravity theories using a masslike function.

## Key findings

- First law of thermodynamics derived from Friedmann equations in Einstein gravity.
- Extension of the thermodynamic-Friedmann connection to Lovelock, nonlinear, and scalar-tensor theories.
- Indicates a fundamental physical relationship beyond coincidence.

## Abstract

With the help of a masslike function which has dimension of energy and equals to the Misner-Sharp mass at the apparent horizon, we show that the first law of thermodynamics of the apparent horizon $dE=T_AdS_A$ can be derived from the Friedmann equation in various theories of gravity, including the Einstein, Lovelock, nonlinear, and scalar-tensor theories. This result strongly suggests that the relationship between the first law of thermodynamics of the apparent horizon and the Friedmann equation is not just a simple coincidence, but rather a more profound physical connection.