Resource Recommendations on The Euclidean Method of Black Hole Thermodynamics I'm trying to derive the thermodynamics of the black hole solution presented at
A New Class of Exact Hairy Black Hole Solutions by Kolyvaris et al.
I'm asking for resource recommendations on this matter, $3+1$ decomposition, boundary terms etc. Is there any paper that derives black hole thermodynamics, using the Euclidean method, for a simpler solution and describes the procedure in detail?
An answer presenting the calculation and procedure in detail is also appreciated!
 A: As a fundamental tool in black hole thermodynamics, the Euclidean method has been applied to both complicated and simple solutions. The sources I would read are the following.

*

*Thermodynamics of black holes in anti-de Sitter space by Hawking and Page. This is the paper which first developed the method and it is very readable. The calculation starts in section 2 and one cannot help but notice how streamlined it is compared to Hawking's earlier tour de force which derived evaporation for black holes in Minkowski space.

*Anti de-Sitter space, thermal phase transition and confinement in gauge theories by Witten. This starts off by reviewing the Hawking Page calculation for AdS black holes and generalizing it to $n$ dimensions. Later on, we get to see it applied in a situation which still receives much attention, obtained by compactiying AdS on a circle in order to give the dual gauge theory a scale.

*Previous SE answers from 2013 and 2017.

As discussed in the latter, a major point of the derivation should already be familiar if we think about converting between polar and Cartesian co-ordinates for Euclidean space.
\begin{equation}
ds^2 = dr^2 + r^2 d\theta^2 \to ds^2 = dx^2 + dy^2
\end{equation}
The polar form shows a vanishing component at $r = 0$. This is shown to be merely a co-ordinate singularity IF we can write it in the Cartesian form. However, writing it in the Cartesian form is only valid if $\theta$ is $2\pi$ periodic because we need to be able to set
\begin{equation}
(x, y) = (r\cos \theta, r\sin \theta).
\end{equation}
In other words,
\begin{equation}
ds^2 = dr^2 + r^2 d\theta^2, \;\;\; \theta \in [0, \alpha]
\end{equation}
is a different manifold from Euclidean space when $\alpha \neq 2\pi$. It is situations like these which show us that the Einstein Hilbert action is a functional which acts on manifolds. Writing it as something which acts on the metric alone is abuse of notation. This is similar to how tori $\mathbb{T}^2$ with different complex structures $z \sim z + 2\pi \tau$ are also different manifolds. Because even though tilting the axes is innocuous in $\mathbb{R}^2$, it becomes meaningful when there are periodicities involved.
Let's see how this works for the Euclidean Schwarzschild metric
\begin{equation}
ds^2 = \left ( 1 - \frac{2M}{r} \right )d\tau^2 + \left ( 1 - \frac{2M}{r} \right )^{-1} dr^2
\end{equation}
by forcing it to pick up a simple $d\rho^2$ term in a new co-ordinate system. This means we need to solve
\begin{equation}
dr = \sqrt{1 - \frac{2M}{r}} d\rho
\end{equation}
which is separable. The solution is
\begin{equation}
\rho = \mathrm{const} + \sqrt{8M(r - 2M)} + O(r - 2M).
\end{equation}
We can now plug this into the metric and again drop higher powers of $r - 2M$ to find
\begin{equation}
ds^2 = d\rho^2 + (\rho / 4M)^2 d\tau^2
\end{equation}
near the horizon. Then in precise analogy with the polar co-ordinate example above, $\tau / 4M$ needs to have period $2\pi$. This implies that the inverse temperature (the period for $\tau$ itself) is given by
\begin{equation}
\beta = 8\pi M
\end{equation}
which is the well known result.
