Solution
= Solution
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
$$
\boxed{\beta_H=
\frac{2\pi L^2r_H}{2r_H^2+L^2}.}
$$