  We study the thermal transport properties of three CaF$_{2}$ polymorphs up to
a pressure of 30 GPa using first-principle calculations and an interatomic
potential based on machine learning. The lattice thermal conductivity $\kappa$
is computed by iteratively solving the linearized Boltzmann transport equation
(BTE) and by taking into account three-phonon scattering. Overall, $\kappa$
increases nearly linearly with pressure, and we show that the recently
discovered $\delta$-phase with $P\bar{6}2m$ symmetry and the previously known
$\gamma$-CaF$_{2}$ high-pressure phase have significantly lower lattice thermal
conductivities than the ambient-thermodynamic cubic fluorite ($Fm\bar{3}m$)
structure. We argue that the lower $\kappa$ of these two high-pressure phases
stems mainly due to a lower contribution of acoustic modes to $\kappa$ as a
result of their small group velocities. We further show that the phonon mean
free paths are very short for the $P\bar{6}2m$ and $Pnma$ structures at high
temperatures, and resort to the Cahill-Pohl model to assess the lower limit of
thermal conductivity in these domains.
