Hawking temperature and the inverse-radius scale of the horizon

The Hawking temperature of a Schwarzschild black hole can be heuristically derived by identifying the temperature with the inverse radius of the horizon up to a multiplicative constant. This does not work for more general black holes such as the Kerr and Reissner–Nordström solutions. Expounding on the details of how it fails to work nevertheless uncovers some connections with the “spring constant” of black holes and with black hole thermodynamics.