In the thermodynamics of the conventional macroscopic systems the entropy is the extensive quantity, i.e. it is proportional to the volume of the system. That is why the splitting of the system with volume $V$ in two parts with volumes $V_1+V_2=V$ does not change the total entropy of the system,  $S(V_1+V_2)=S(V_1) +S(V_2)$. This, however, is not applicable to the thermodynamics of the black hole: the entropy of the black hole is not extensive. The splitting of the black hole of mass $M$ to two black holes with the same total mass $M_1+M_2=M$ leads to the decrease of the total entropy,  $S(M_1) +S(M_2) < S(M_1+M_2)$.

The composition rule for the black holes and thus the black hole thermodynamics can be obtained by consideration of the quantum processes -- processes of quantum tunneling. The calculations of the rate of quantum tunneling process of radiation of particles and photons from the black hole hole gives the temperature of the black hole radiation, which coincides with the Hawking temperature. On the other hand, the radiation of small black holes by the larger ones (or in general the splitting of the black hole to the smaller parts) are described by the processes of macroscopic quantum tunneling.

The macroscopic quantum tunneling is the rear event caused by quantum fluctuations, which in the black hole thermodynamics can be considered in the same manner as thermal fluctuations.
According to Landau-Lifshitz book "Statistical Physics", thermal fluctuations in the macroscopic system lead to decrease of the entropy, and thus the processes of the black hole fragmentation are determined by the change of the entropy after the splitting.  Then, the calculations of the rate of the fragmentation using the method of the macroscopic quantum tunneling allow us to obtain the composition rule for the black hole entropy:
\begin{equation}
\sqrt{S(M_1 + M_2)}=\sqrt{S(M_1)}+ \sqrt{S(M_2)}\,,
\label{CompositionN}
\end{equation}
which reproduces the Bekenstein-Hawking entropy. So, the macroscopic quantum tunneling approach is actually another way for the derivation of the black hole entropy.
 
 This composition rule demonstrates that the black holes obey the special type of the generalized statistics -- the Tsallis-Cirto statistics with $\delta=2$. This also suggests that the black hole  can be considered as an ensemble of the elementary black holes with the Planck-scale mass $M_0=1/\sqrt{4\pi G}$. The black hole mass is $M=NM_0$, where $N$  is the number of these micro black holes, which play the role of the black hole "atoms". The  Bekenstein–Hawking entropy of the black hole with $N$ atoms, is $S(N)=N^2$.

G.E. Volovik
JETP Letters 121, issue 4 (2025)