With the detection of GW150914, the era of gravitational wave astronomy has commenced. One possible source of gravitational waves is accreting neutron stars. Many low mass X-ray binary neutron stars are spinning at frequencies considerably lower than the neutron star break-up frequency. Gravitational wave emission might account for this observed maximum spin cap. For an isolated neutron star to emit gravitational waves, it must deform from its axial symmetry to produce a time-varying gravitational field. One way this can occur is through the development of a misaligned quadrupole moment. A quadrupole moment or ‘mountain’ can develop if temperature asymmetries exist in a neutron star crust.In this thesis, we investigate whether temperature asymmetries can develop in an accreted neutron star crust. We construct a self-consistent model of a spherically symmetric background thermal profile of an accreted crust. A temperature perturbation is then induced by inserting a magnetic field. The presence of a magnetic field causes anisotropies in the thermal conductivity to develop, due to electrons interacting with the field. We explore the parameter space of accretion rate, impurity parameter and magnetic field strength. We then investigate the influence of shallow crustal heating on our model. Later, we consider the effects of existing temperature asymmetries on the surface of the crust, which can arise from non-spherical accretion. We find these perturbation mechanisms are unlikely to induce temperature asymmetries that can produce a sufficiently large mass quadrupole moment which generates energy losses via gravitational wave emission to balance the spin-up torque from accretion.