Boltzmann factor and Hawking radiation

Hawking radiation has thermal spectrum corresponding to the temperature $T_H={(8\pi M)}^{-1}$, where $M$ is the mass (energy) of the black hole. Corrections to the Hawking radiation spectrum were discovered by Kraus and Wilczek (1995) and Parikh and Wilczek (2000). Here I show that these corrections follow directly from the basic principles of thermodynamics and statistical mechanics. In essence, it is the Boltzmann factor that ought to be corrected; corrections to the Hawking (or any other) radiation spectrum then follow necessarily.