Let’s see if we just choose the first part in \eqref{eq:bw-s}, how the entropy looks like, and it’s meanings. Schwarzschild black hole’s surface gravity is

\kappa = \frac{1}{4M}, \\

and $\beta = \frac{1}{T_H}=\frac{2\pi}{\kappa}=8\pi M$, $r_H = 2M$ is the event horizon, then the entropy is

S=\frac{2M}{360h}.

If we choose the cutoff as
\label{eq:bw-cutoff}
h=\frac{T_H}{90}=\frac{1}{720\pi M}=\frac{1}{90\beta},

then the entropy becomes

S=4\pi M^2=\pi r_H^2=\frac{1}{4}A.

where $A=4\pi r_H^2$ is the area of the event horizon. The entropy is the Bekenstein-Hawking entropy.