Conceptually, why is the entropy of black hole related to Planck length? I was watching a lecture on the holographic principle, and it presented the equation for the entropy of a black hole as $$S_{BH} = \frac{A}{4l_p^2}$$ My question is why, conceptually, the entropy of a black hole should be related to Planck length?
 A: In classical general relativity there is no deep explanation, since we don't consider quantum mechanics. The appearance of the Planck length is due to the Geometrized unit system commonly used in general relativity. Since the area has the dimensionality of a squared length we need a factor in the denominator with squared length dimensions, so that the entropy is dimensionless. The Planck length is usually chosen as a unit of reference, since it is naturally obtained without any multiplicative factor from the combination of common constants such as $c$, $G$ and $\hbar$. Indeed the Planck length is central to other systems of units, such as the Planck units.
The real question is why does the Planck length arise with a $O(1)$ factor in the formula?
One can make many arguments such as that the Planck length arises due to:

*

*The quantum nature of the underlying statistical microstates. Original Bekenstein quote:


The appearance of $\hbar$ is “... a reflection of the fact that the entropy
is ... a count of states of the system, and the underlying states are
quantum in nature ... . It would be somewhat pretentious to calculate
the precise value of η without a full understanding of the quantum
reality which underlies a ‘classical’ black hole”


*

*Black holes evaporation and thermodynamic nature, similarly to a regular black body radiation.


*Quantum gravity holographic nature of the black hole, as suggested by the presence of the area.


*The relation to the entanglement entropy of quantum fields and quantum extremal surfaces (see Ryu–Takayanagi conjecture)
But to my knowledge we still lack a convincing and deep argument about the (1) factor. We need a better understanding of the quantum theory of which gravity is a coarse grained picture.
