Thermodynamics of Taub-NUT/Bolt-AdS Black Holes in Einstein-Gauss-Bonnet Gravity

We give a review of the existence of Taub-NUT/bolt solutions in Einstein Gauss-Bonnet gravity with the parameter $\alpha $ in six dimensions. Although the spacetime with base space $S^{2}\times S^{2}$ has curvature singularity at $r=N$, which does not admit NUT solutions, we may proceed with the same computations as in the $\mathbb{CP}^{2}$ case. The investigation of thermodynamics of NUT/Bolt solutions in six dimensions is carried out. We compute the finite action, mass, entropy, and temperature of the black hole. Then the validity of the first law of thermodynamics is demonstrated. It is shown that in NUT solutions all thermodynamic quantities for both base spaces are related to each other by substituting $\alpha^{\mathbb{CP}^{k}}=[(k+1)/k]\alpha^{S^{2} \times S^{2}\times >...S_{k}^{2}}$. So no further information is given by investigating NUT solution in the $S^{2}\times S^{2}$ case. This relation is not true for bolt solutions. A generalization of the thermodynamics of black holes to arbitrary even dimensions is made using a new method based on the Gibbs-Duhem relation and Gibbs free energy for NUT solutions. According to this method, the finite action in Einstein Gauss-Bonnet is obtained by considering the generalized finite action in Einstein gravity with an additional term as a function of $\alpha$. Stability analysis is done by investigating the heat capacity and entropy in the allowed range of $\alpha$, $\Lambda$ and $N$. For NUT solutions in $d$ dimensions, there exist a stable phase at a narrow range of $\alpha$. In six-dimensional Bolt solutions, metric is completely stable for $\mathcal{B}=S^{2}\times S^{2}$, and is completely unstable for $\mathcal{B}=\mathbb{CP}^{2}$ case.