= Solution
In natural units, periodic Euclidean time of period beta is the inverse <temperature> of a thermal quantum state. The horizon regularity calculation therefore gives the <Hawking temperature>
$$
\boxed{T_H=\frac1{\beta_+}=\frac{V'(r_+)}{4\pi}=\frac{\kappa_+}{2\pi}=\frac{\sqrt{M^2-Q^2}}{\pi r_+^3}.}
$$
This agrees with the radiation calculation based on the near-horizon redshift. Restoring constants with $c=1$ multiplies the expression $\kappa_+/(2\pi)$ by $\hbar/k_B$. In the neutral case $T_H=1/(2\pi\sqrt{2M})$; the <temperature> is positive for every nonextremal outer horizon.
The extremal limit is different. At $|Q|=M$, $V\sim4(r-r_h)^2/M$, so the proper Euclidean radial distance to the horizon diverges logarithmically. This <extremal Euclidean throat> is an infinite end, with no finite polar origin on which to impose the cone-removal condition. Local smoothness therefore does not select a finite period by the preceding argument. The vanishing <surface gravity> and the nonextremal limit give $\boxed{T_H=0\text{ at extremality}}$, rather than a universally nonzero <temperature> for all horizon solutions. A geometry with $|Q|>M$ has no black-hole horizon and no <Hawking temperature> inferred from this horizon argument.
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