  We report on the temperature dependence of the dielectric tensor elements of
$n$-type conductive $\beta$-Ga$_2$O$_3$ from 22$^\circ$C-500$^\circ$C in the
spectral range of 1.5~eV--6.4~eV. We present the temperature dependence of the
excitonic and band-to-band transition energies and their eigenpolarization
vector orientations. We utilize a Bose-Einstein analysis of the temperature
dependence of the observed transition energies and reveal electron coupling
with average phonon temperature in excellent agreement with the average over
all longitudinal phonon plasmon coupled modes reported previously [M.
Schubert~\textit{et al.}, Phys. Rev. B \textbf{93}, 125209 (2016)]. We also
report a linear temperature dependence of the wavelength independent Cauchy
expansion coefficient for the anisotropic below-band-gap monoclinic indices of
refraction.
