Let $R=R_{\rm AdS}$ and $r_c=\pi/2-\epsilon$. For $\operatorname{AdS}_3$, $\mathcal R=-6/R^2$ and $\Lambda=-1/R^2$, while
$$
\sqrt{-g}=R^3\frac{\sin r}{\cos^3r}.
$$
The bulk term is
$$
I_{\rm bulk}=-\frac{R\Delta t}{4G}(\sec^2r_c-1).
$$
With the inward normal $n^r=-\cos r/R$, the induced metric has $\sqrt{-h}=R^2\sin r/\cos^2r$ and
$$
K=-\frac1R\left(\frac{\cos^2r}{\sin r}+2\sin r\right).
$$
The <Gibbons–Hawking–York boundary term> is consequently
$$
I_{\rm GHY}=\frac{R\Delta t}{4G}(1+2\tan^2r_c).
$$
Thus, with the orientation and Lorentzian signs displayed in the question,
$$
\boxed{I_{\rm GR}=\frac{R\Delta t}{4G}\sec^2r_c
=\frac{R\Delta t}{4G}\csc^2\epsilon.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 354 / 2 / b / i / Solution

Keep the smooth $\beta=\beta_H$ geometry fixed away from the horizon but identify Euclidean time with arbitrary period $\beta$. The horizon then has deficit angle $2\pi(1-\beta/\beta_H)$. Its delta-function curvature contributes
$$
\int_{\mathcal M}\sqrt g\,R\big|_{\rm tip}
=4\pi\left(1-\frac\beta{\beta_H}\right)A,
$$
and hence
$$
I_{\rm tip}=-\frac{A}{4G_N}
\left(1-\frac\beta{\beta_H}\right).
$$
Using $\log Z=-I_{\rm grav}$,
$$
\boxed{S_{\rm BH}
=\left.(1-\beta\partial_\beta)\log Z\right|_{\beta_H}
=\frac{A}{4G_N}
=\frac{2\pi^2r_H^3}{4G_N}.}
$$

All smooth bulk terms, including the cosmological-constant volume term, are proportional to the Euclidean time period and are annihilated by $1-\beta\partial_\beta$. The asymptotic <Gibbons–Hawking–York boundary term> and holographic counterterms are likewise smooth and linear in $\beta$. Only the curvature singularity at the fixed point of the Euclidean time circle survives.

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 2 / b / i / Solution

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The Gibbons–Hawking–York boundary term makes the Einstein--Hilbert variational problem well posed when the induced boundary metric is fixed. It is the integral of the trace of the boundary's extrinsic curvature with the conventional normalization and sign.


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