  Context. Large thermal variations have been observed in neutron stars that
typically are not aligned with density gradients. Such terms may activate the
Biermann battery effect, leading to thermoelectric interactions and to the
generation of electromotive force. Aims. We aim to identify the possible impact
of a temperature anisotropy on the crust of a neutron star can have in the
evolution of its magnetic field, through the thermoelectric terms. Methods. We
consider a neutron star crust with large temperature gradients, associated with
long-lived hot spots, described by a localized Gaussian-type function. We
simulate the interplay between the battery term and the Hall and Ohmic
evolution numerically for axisymmetric systems. Results. The results indicate
that for crust temperatures of $\sim$$10^9$ K the toroidal field can be
amplified up to $\sim$$10^{14}-10^{15}$ G near the points of maximum
temperature gradients, and it locally changes the architecture of the poloidal
field lines. For internal crustal temperatures of $\sim$$10^8$ K, the
temperature gradient generates fields that are lower by about two orders of
magnitude. In these cases, saturation is achieved after some hundred thousand
years, after which the battery and Ohmic dissipation balance each other,
whereas the Hall drift contributes comparatively little to the final field
strength, but it can affect its structure. Conclusions. We conclude that the
thermoelectric effect can impact the overall magnetic field evolution, provided
that the thermal gradient is maintained for a sufficiently long time. Neutron
stars endowned with moderate-strength magnetic fields may be affected by the
thermoelectric effect if the hotspots survive for timescales of a few
kiloyears.
