= Solution
Classically, the <laws of thermodynamics> and <black-hole thermodynamics> have parallel structures, but an ordinary black hole absorbs and does not emit: its temperature would appear to be zero, despite nonzero <surface gravity>. <Hawking radiation> changes this by giving an actual temperature measurable from a thermal particle spectrum. In units $G=\hbar=c=k_B=1$,
$$
\boxed{T_H=\frac\kappa{2\pi}.}
$$
Scattering through the exterior modifies the observed flux by <greybody factors>; for rotation and charge the occupation involves $\omega-m_\phi\Omega_H-q\Phi_H$. These transmission and chemical-potential factors do not change the underlying Hawking temperature.
The <Zeroth law of black-hole mechanics> makes $\kappa$ constant on a stationary horizon, so it gives a uniform equilibrium temperature, matching the <zeroth law of thermodynamics>. The full <First law of black-hole mechanics> is
$$
dM=\frac\kappa{8\pi}\,dA+\Omega_H\,dJ+\Phi_H\,dQ.
$$
Comparing it with $dE=T\,dS+\Omega\,dJ+\Phi\,dQ$ and using the actual Hawking temperature fixes
$$
\boxed{S_{BH}=\frac A4,\qquad
dM=T_H\,dS_{BH}+\Omega_H\,dJ+\Phi_H\,dQ.}
$$
With constants restored, $S_{BH}=k_Bc^3A/(4G\hbar)$. If $\kappa$ is defined geometrically with dimensions of inverse length, $T_H=\hbar c\kappa/(2\pi k_B)$; if surface gravity is expressed as an acceleration, the equivalent formula is $\hbar\kappa/(2\pi c k_B)$. The first-law comparison fixes entropy up to an additive constant. This is the <Bekenstein-Hawking entropy>, so the mechanical analogy becomes a quantitative thermodynamic identification, including its normalization.
The classical <second law of black-hole mechanics> then corresponds to increasing black-hole entropy. Quantum evaporation can reduce $A$, because the classical energy-condition hypotheses fail, while outgoing radiation carries entropy. The appropriate thermodynamic statement is the <generalized second law>,
$$
\boxed{S_{\rm gen}=S_{BH}+S_{\rm outside}\text{ does not decrease}.}
$$
Finally, the <third law of black-hole mechanics>, that zero surface gravity cannot be achieved by a finite physical process under its regularity assumptions, becomes the unattainability of zero temperature. It does not assert the Nernst entropy formulation: an extremal horizon can have nonzero area. Hawking radiation supplies the physical temperature, the first law fixes entropy, and the generalized second law includes both the black hole and its exterior matter.
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