By means of first-principles calculations, we investigate the thermal properties of silica as it evolves, under hydrostatic compression, from a stishovite phase into a CaCl$_2$-type structure. We compute the thermal conductivity tensor by solving the linearized Boltzmann transport equation iteratively in a wide temperature range, using for this the pressure-dependent harmonic and anharmonic interatomic couplings obtained from first principles. Most remarkably, we find that, at low temperatures, SiO$_2$ displays a large peak in the in-plane thermal conductivity and a highly anisotropic behavior close to the structural transformation. We trace back the origin of these features by analyzing the phonon contributions to the conductivity. We discuss the implications of our results in the general context of continuous structural transformations in solids, as well as the potential geological interest of our results for silica.
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