After Wick rotation $t=-it_E$, the near-horizon metric has
$$
ds_E^2\simeq f'(r_H)(r-r_H)dt_E^2
+\frac{dr^2}{f'(r_H)(r-r_H)}.
$$
Regularity at the origin of this polar plane requires the <Euclidean black-hole regularity condition> $\beta_H=4\pi/f'(r_H)$. Since
$$
\mu=r_H^2\left(1+\frac{r_H^2}{L^2}\right),
\qquad
f'(r_H)=\frac2{r_H}+\frac{4r_H}{L^2},
$$
we obtain
$$
\boxed{\beta_H=
\frac{2\pi L^2r_H}{2r_H^2+L^2}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 2 / a / ii / Solution

Near a nonextremal Euclidean horizon, the radial--time plane is smooth only when Euclidean time has period $\beta_H=4\pi/f'(r_H)$. Any other period produces a conical singularity at the horizon.


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 Incoming links: Euclidean black-hole regularity condition