Bekenstein-Hawking entropy ($S_{BH}$)

In units $G=\hslash =c=k_B=1$, a black hole of horizon area $A$ has entropy $S_{BH}=A/4$.

Changing the Euclidean time period away from its smooth value creates a curvature delta function at the horizon. Applying $(1-\beta {\partial }_{\beta })$ to the Euclidean gravitational partition function isolates this term and gives $S=A/(4G_N)$.