= Solution
Classically, the <laws of black-hole mechanics> have the same form as the laws of <thermodynamics>, but this could have been only a formal analogy. <Hawking radiation> makes it physical: a stationary black hole emits approximately <thermal radiation> with
$$
\boxed{T_H=\frac{\hbar\kappa}{2\pi k_Bc}},
$$
so <surface gravity> is proportional to genuine <temperature>.
The correspondence is:
* constancy of $\kappa$ on a stationary horizon becomes constancy of equilibrium temperature;
* the <First law of black-hole mechanics>
$$
dM=\frac{\kappa}{8\pi G}\,dA+\Omega_H\,dJ+\Phi_H\,dQ
$$
matches the <first law of thermodynamics> and fixes the <Bekenstein-Hawking entropy>
$$
\boxed{S_{BH}=\frac{k_Bc^3A}{4G\hbar}};
$$
* <Hawking's area theorem> becomes a second law, replaced in the quantum theory by the <generalized second law> for $S_{BH}+S_{\rm outside}$;
* the impossibility of reaching $\kappa=0$ by a finite process parallels the unattainability of absolute zero.
A real Hawking temperature therefore identifies horizon area as thermodynamic entropy and turns the geometric laws into thermodynamic laws.
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