  A magnetically confined mountain on the surface of an accreting neutron star
simultaneously reduces the global magnetic dipole moment through magnetic
burial and generates a mass quadrupole moment, which emits gravitational
radiation. Previous mountain models have been calculated for idealized
isothermal and adiabatic equations of state. Here these models are generalised
to include non-zero, finite thermal conduction. Grad-Shafranov equilibria for
three representative, polytropic equations of state are evolved over many
conduction time-scales with the magnetohydrodynamic solver PLUTO. It is shown
that conduction facilitates the flow of matter towards the pole. Consequently
the buried magnetic field is partially resurrected starting from an initially
polytropic Grad-Shafranov equilibrium. The poleward mass current makes the star
more prolate, marginally increasing its detectability as a gravitational wave
source, though to an extent which is likely to be subordinate to other mountain
physics. Thermal currents also generate filamentary hot spots $(\gtrsim 10^{8}
\text{ K})$ in the mountain, especially near the pole where the heat flux is
largest, with implications for type I X-ray bursts.
