Static pure Lovelock black hole solutions with horizon topology ${\bf S^{(n)} \times S^{(n)}}$

TL;DR

This paper derives static black hole solutions with ${f S^{(n)} imes S^{(n)}}$ topology in pure Lovelock gravity of arbitrary order, revealing stability differences based on the order's parity and cosmological constant sign.

Contribution

It extends pure Lovelock black hole solutions to arbitrary order with two-sphere topology and analyzes their thermodynamic stability.

Findings

Thermodynamical stability distinguishes between odd and even Lovelock orders.

Stability favors negative cosmological constant for these solutions.

Explicit solutions are obtained for third and fourth order Lovelock gravity.

Abstract

It is well known that vacuum equation of arbitrary Lovelock order for static spacetime ultimately reduces to a single algebraic equation, we show that the same continues to hold true for pure Lovelock gravity of arbitrary order $N$ for topology ${\bf S^{(n)} \times S^{(n)}}$. We thus obtain pure Lovelock static black hole solutions with two-sphere topology for any order $N$, and in particular we study in full detail the third and fourth order Lovelock black holes. It is remarkable that thermodynamical stability of black hole discerns between odd and even $N$, and consequently between negative and positive $\Lambda$ and it favours the former while rejecting the latter.