# The extended uncertainty principle inspires the R\'{e}nyi entropy

## TL;DR

This paper derives the Rényi entropy for black holes using the extended uncertainty principle, linking non-extensivity to quantum states, and explores thermodynamic properties and evaporation times of excited black holes.

## Contribution

It introduces a novel approach connecting EUP to Rényi entropy for black holes and analyzes thermodynamic behavior of excited states.

## Key findings

- Rényi entropy depends on black hole quantum states.
- Temperature can have a positive minimum even when entropy vanishes.
- Evaporation time of excited black holes is characterized.

## Abstract

We use the extended uncertainty principle (EUP) in order to obtain the R\'{e}nyi entropy for a black hole (BH). The result implies that the non-extensivity parameter, appeared in the R\'{e}nyi entropy formalism, may be evaluated from the considerations which lead to EUP. It is also shown that, for excited BHs, the R\'{e}nyi entropy is a function of the BH principal quantum number, i.e. the BH quantum excited state. Temperature and heat capacity of the excited BHs are also investigated addressing two phases while only one of them can be stable. At this situation, whereas entropy is vanished, temperature may take a non-zero positive minimum value, depending on the value of the non-extensivity parameter. The evaporation time of excited BH has also been studied.