Euclidean metric near a simple static horizon ($d{s}_E^2\simeq d{\rho }^2+\frac{1}{4}{V}^{\prime }(r_0{)}^2{\rho }^{2}d{\tau }^2$)

On the side where V is positive, set $r-r_0=V^{\prime }(r_0){\rho }^2/4$. Its radial Euclidean metric becomes the displayed plane in polar coordinates. The would-be singularity is conical unless the angular variable $|V^{\prime }(r_0)|\tau /2$ has period exactly $2\pi$. This fixes $\beta =4\pi /|V^{\prime }(r_0)|$; a nontrivial integer multiple still has a cone excess at the tip.