# Action Growth in $f\left(R\right)$ Gravity

## TL;DR

This paper investigates the rate of action growth in $f(R)$ gravity black holes within the 'Complexity = Action' framework, revealing bounds' saturation and violation depending on charge and dimensions.

## Contribution

It extends the 'Complexity = Action' conjecture to $f(R)$ gravity, analyzing action growth and Lloyd bound behavior for various black hole charges and dimensions.

## Key findings

- Lloyd bound is saturated for uncharged black holes.
- Lloyd bound is violated for charged black holes near extremality.
- Violation of Lloyd bound depends on charge and spacetime dimensions.

## Abstract

Inspired by the recent "Complexity = Action" conjecture, we use the approach proposed by Lehner et al. to calculate the rate of the action of the WheelerDeWitt patch at late times for static uncharged and charged black holes in $f\left( R\right) $ gravity. Our results have the same expressions in terms of the mass, charge, and electrical potentials at the horizons of black holes as in Einstein's gravity. In the context of $f\left( R\right) $ gravity, the Lloyd bound is saturated for uncharged black holes but violated for charged black holes near extremality. For charged black holes far away from the ground states, the Lloyd bound is violated in four dimensions but satisfied in higher dimensions.