  Bearing in mind the application to high-magnetic-field (high-B) radio
pulsars, we investigate two-dimensional (2D) thermal evolutions of neutron
stars (NSs). We pay particular attention to the influence of different
equilibrium configurations on the surface temperature distributions. The
equilibrium configurations are constructed in a systematic manner, in which
both toroidal and poloidal magnetic fields are determined self-consistently
with the inclusion of general relativistic effects. To solve the 2D heat
transfer inside the NS interior out to the crust, we have developed an implicit
code based on a finite-difference scheme that deals with anisotropic thermal
conductivity and relevant cooling processes in the context of a standard
cooling scenario. In agreement with previous studies, the surface temperatures
near the pole become higher than those in the vicinity of the equator as a
result of anisotropic heat transfer. Our results show that the ratio of the
highest to the lowest surface temperatures changes maximally by one order of
magnitude, depending on the equilibrium configurations. Despite such
difference, we find that the area of such hot and cold spots is so small that
the simulated X-ray spectrum could be well reproduced by a single temperature
blackbody fitting.
