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 \begin{equation} \kappa = \frac{1}{4M}, \end{equation} and $\beta = \frac{1}{T_H}=\frac{2\pi}{\kappa}=8\pi M $, $r_H = 2M $ is the event horizon, then the entropy is \begin{equation} S=\frac{2M}{360h}. \end{equation} If we choose the cutoff as \begin{equation}\label{eq:bw-cutoff} h=\frac{T_H}{90}=\frac{1}{720\pi M}=\frac{1}{90\beta}, \end{equation} then the entropy becomes \begin{equation} S=4\pi M^2=\pi r_H^2=\frac{1}{4}A. \end{equation} where \(A=4\pi r_H^2\) is the area of the event horizon. The entropy is the Bekenstein-Hawking entropy.