= Solution
Use $G=\hbar=c_{\rm light}=k_B=1$ and normalize the stationary time at infinity. For a regular stationary, asymptotically flat electrovacuum <black hole>, the <Kerr-Newman metric> has
$$
a=\frac JM,\qquad d=\sqrt{M^2-Q^2-a^2},\qquad
r_\pm=M\pm d,\qquad
D_H=r_+^2+a^2=2M^2-Q^2+2Md.
$$
Initially assume $M>0$ and $d>0$, so the outer <Killing horizon> is nondegenerate. Its <surface gravity>, angular velocity, electric potential and area are
$$
\kappa=\frac{r_+-r_-}{2D_H}=\frac d{D_H},\qquad
\Omega_H=\frac a{D_H},\qquad
\Phi_H=\frac{Qr_+}{D_H},\qquad
A_H=4\pi D_H.
$$
These are horizon quantities for the generator $\chi=\partial_t+\Omega_H\partial_\phi$.
The <laws of black-hole mechanics> provide the first reason to regard $\kappa$ and area as thermodynamic variables. The <Zeroth law of black-hole mechanics> makes $\kappa$ constant on an equilibrium horizon. The <first law for the Kerr-Newman family> reads
$$
dM=\frac{\kappa}{8\pi}\,dA_H+\Omega_H\,dJ+\Phi_H\,dQ.
$$
For example, write the horizon relation as $M=(r_+^2+a^2+Q^2)/(2r_+)$ and use $J=Ma$; differentiating eliminates $dr_+$ and $da$ to give this law. The last two terms are rotational and electric work. The classical <second law of black-hole mechanics> says that area cannot decrease under the <null energy condition> and appropriate global horizon assumptions. The <third law of black-hole mechanics> concerns unattainability of a zero-surface-gravity regular horizon by a finite physical process.
This analogy alone does not determine the <temperature> scale: if entropy were $\eta A_H$, the first law would only give $T=\kappa/(8\pi\eta)$. The decisive input is <quantum field theory> in the collapsing or stationary background. In a collapse vacuum that is regular for freely falling observers, the late outgoing retarded time $u$ and a regular affine null coordinate $U$ satisfy the <Hawking exponential ray map>
$$
U=-C e^{-\kappa u},\qquad C>0.
$$
An outgoing mode $e^{-i\omega u}$ therefore behaves as $(-U/C)^{i\omega/\kappa}$. Continuing this power through the horizon changes its amplitude by the factor $e^{-\pi\omega/\kappa}$. Its positive- and negative-frequency decomposition consequently has the <thermal ratio of Hawking Bogoliubov coefficients>
$$
\frac{|\beta_\omega|^2}{|\alpha_\omega|^2}=e^{-2\pi\omega/\kappa}.
$$
Combining that ratio with bosonic normalization $|\alpha|^2-|\beta|^2=1$ gives a Planck occupation $1/(e^{2\pi\omega/\kappa}-1)$. The fermionic normalization instead gives the corresponding Fermi factor. Thus <Hawking radiation> fixes
$$
T_H=\frac{\kappa}{2\pi}.
$$
For rotating charged modes, the horizon energy is $\widetilde\omega=\omega-m_\phi\Omega_H-q\Phi_H$, so the emission spectrum contains the same <temperature> with angular-momentum and charge chemical potentials. Exterior scattering supplies <greybody factors>; it changes the received flux, not the horizon <temperature>. Superradiant bosonic modes require the usual signed absorption factor, and a globally regular thermal bath need not exist throughout an asymptotically flat rotating exterior. The collapse-state emission argument is the relevant one.
A complementary check is the <Euclidean black-hole regularity condition>. The local corotating nondegenerate horizon geometry is Rindler-like:
$$
ds_E^2\simeq d\rho^2+\kappa^2\rho^2d\tau^2+\text{horizon metric}.
$$
Smoothness at $\rho=0$ requires $\tau$ to have period $2\pi/\kappa$. Imaginary-time periodicity is precisely inverse <temperature>, agreeing with the radiation calculation. Matching the resulting <temperature> to the first law fixes the <Bekenstein-Hawking entropy> to $S=A_H/4$. The <generalized second law> then uses $S_{\rm outside}+A_H/4$: the classical area theorem alone does not apply to the negative-energy quantum flux responsible for evaporation.
Substitution gives the <Kerr-Newman horizon temperature>
$$
\boxed{T_H=\frac{\sqrt{M^2-Q^2-J^2/M^2}}
{2\pi\left(2M^2-Q^2+2M\sqrt{M^2-Q^2-J^2/M^2}\right)}.}
$$
It reduces to $1/(8\pi M)$ for a <Schwarzschild black hole>. The nonextremal limit towards $d=0$ gives zero <temperature>. At exact extremality the Euclidean horizon is degenerate, so the elementary conical-period argument does not itself fix a period; zero <temperature> here is the limiting semiclassical result. The reasoning assumes ordinary Einstein–Maxwell dynamics, a regular horizon, the stated normalization at infinity and a regime where quantum fields on a slowly evolving classical geometry are a useful approximation. The mechanical laws, radiation spectrum and Euclidean regularity agree under these assumptions, which is substantially stronger evidence than the classical analogy alone.
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