  Photoluminescence (PL) is a light matter quantum interaction associated with
the chemical potential of light formulated by the Generalized Planck's law.
Without knowing the inherent temperature dependence of chemical potential, the
Generalized Planck's law is insufficient in order to characterize PL vs. T.
Recent experiments showed that PL at a critical temperature abruptly shifts
from a conserved rate, accompanied by a blue-shift, to thermal emission. Here,
we theoretically study temperature dependent PL by including phononic
interactions in a detailed balance analysis. We show that in a three-level
system, both chemical potential and T are defined in the case of fast
thermalization. Our solution validates recent experiments and predicts new
features, including an inherent relation between emissivity and external
quantum efficiency of a system, a universal point defined by the pump and the
temperature where the emission rate is fixed to any material, a new phonon
induced quenching mechanism, and thermalization of the photon spectrum. Our
high temperature luminescence solution is relevant to and important for all
photonic fields where the temperature is dominant
