  We present the first large-scale 3D map of interstellar dust temperature. We
build upon existing 3D reddening maps derived from starlight absorption
(Bayestar19), covering 3/4 of the sky. Starting with the column density for
each of 500 million 3D voxels, we propose a temperature and emissivity
power-law slope ($\beta$) for each of them, and integrate along the line of
sight to synthesize an emission map in five frequency bands observed by
\emph{Planck} and \emph{IRAS}. The reconstructed emission map is constrained to
match observations on a $10'$ scale, and does so with good fidelity. We produce
3D temperature maps at resolutions of $110', 55', $and $27'$. We assess
performance on Cepheus, a dust cloud with two distinct components along the
line of sight, and find distinct temperatures for the two components. We thus
show that this methodology has enough precision to constrain clouds with
different temperature along the line of sight up to $1-\sigma$ error. This
would be an important result for dust frequency decorrelation foreground
analysis for cosmic microwave background experiments, which would be impacted
by a line-of-sight with varying temperature and magnetic field components. In
addition to $T$ and $\beta$, we constrain the conversion factor between
emission optical depth and reddening. This conversion factor is assumed to be
constant in commonly used emission-based reddening maps. However, this work
shows a factor of two variation that may prove significant for some
applications.
