= Solution
The <laws of black-hole mechanics> initially relate geometric quantities in a way resembling <thermodynamics>. <Hawking radiation> supplies a physical temperature: in units $c=1$, while displaying $G,\hbar,k_B$,
$$
\boxed{T_H=\frac{\hbar\kappa}{2\pi k_B},\qquad S_{\rm BH}=\frac{k_B A}{4G\hbar}.}
$$
The <Hawking temperature> $T_H$ is measured with the stationary time normalized at infinity. The radiation's thermal occupation factor is physically observable; propagation to infinity also introduces <greybody factors>, so the distant spectrum need not be a perfect blackbody spectrum at every frequency.
The <Zeroth law of black-hole mechanics> says that <surface gravity> $\kappa$ is constant on a stationary <Killing horizon> under its standard hypotheses. Through the <Hawking temperature>, this becomes uniform equilibrium temperature. The <First law of black-hole mechanics> is
$$
\delta M=\frac\kappa{8\pi G}\delta A+\Omega_H\delta J+\Phi_H\delta Q.
$$
Using $T_H$ identifies its area term as $T_H\delta S_{\rm BH}$. Integrating $\delta S_{\rm BH}=k_B\delta A/(4G\hbar)$ gives the <Bekenstein-Hawking entropy>, up to an additive constant. Thus $M$ is the energy, the angular and charge terms are work terms, and the geometrical law is the ordinary thermodynamic first law with a fixed entropy normalization.
The <second law of black-hole mechanics>, or <Hawking's area theorem>, gives nondecreasing area in the classical setting with the requisite energy and predictability assumptions. It corresponds to increasing <Bekenstein-Hawking entropy>. Semiclassical <black-hole evaporation> can decrease the area: the classical <null energy condition> need not hold for the quantum expectation of the <stress-energy tensor>. The appropriate extension is the <generalized second law>, that $S_{\rm BH}+S_{\rm outside}$ does not decrease, with the exterior entropy and its renormalization treated consistently. Hawking's temperature identification motivates this law; thermality alone does not prove every form of it.
The <third law of black-hole mechanics> is the unattainability, by an admissible finite physical process, of zero <surface gravity>. With <Hawking temperature> it becomes unattainability of absolute zero. It is not the assertion that an <extremal black hole> has vanishing entropy: its area can remain nonzero when $\kappa=0$.
For a <Schwarzschild black hole>, $\kappa=1/(4GM)$ and $A=16\pi G^2M^2$, giving $T_H=\hbar/(8\pi k_BGM)$ and $S_{\rm BH}=4\pi k_BGM^2/\hbar$. Their product satisfies $T_H\,dS_{\rm BH}=dM$, explicitly checking the first law. \b[Quantum radiation turns the temperature and entropy in the mechanical analogy into physical thermodynamic quantities.]
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