  Kelly and Leff demonstrated and discussed formal and conceptual similarities
between basic thermodynamic formulas for the classical ideal gas and black body
photon gas. Leff pointed out that thermodynamic formulas for the photon gas
cannot be deduced completely by thermodynamic methods since these formulas hold
two characteristic parameters, {\it r} and {\it b}, whose accurate values can
be obtained exclusively by accurate methods of the quantum statistics (by
explicit use of the Planck's or Bose-Einstein distribution). In this work we
prove that the complete quantum thermodynamics of the black body photon gas can
be done by simple, thermodynamic (non-statistical) methods. We prove that both
mentioned parameters and corresponding variables (photons number and pressure)
can be obtained very simply and practically exactly (with relative error about
few percent), by non-statistical (without any use of the Planck's or
Bose-Einstein distribution), quantum thermodynamic methods. Corner-stone of
these methods represents a quantum thermodynamic stability condition that is,
in some degree, very similar to quantum stability condition in the Bohr quantum
atomic theory (de Broglie's interpretation of the Bohr quantization postulate).
Finally, we discuss conceptual similarities between black body photon gas
entropy and Bekenstein-Hawking black hole entropy.
