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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 temperatureThis agrees with the radiation calculation based on the near-horizon redshift. Restoring constants with $c=1$ multiplies the expression ${\kappa }_{+}/(2\pi )$ by $\hslash /k_B$. 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 $|Q|=M$, $V\sim 4(r-r_h{)}^2/M$, 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.