  In this work we revisit the thermal relaxation process for neutron stars.
Such process is associated with the thermal coupling between the core and the
crust of neutron stars. The thermal relaxation, which takes place at around 10
-- 100 years, is manifested as a sudden drop of the star's surface temperature.
Such drop is smooth for slow cooling objects and very sharp for those with fast
cooling. In our study we focus particularly on the cooling of neutron stars
whose mass is slightly greater than the value above which the direct Urca (DU)
process sets in. Considering different mechanisms for neutrino production in
each region of the star, and working with equations of state with different
properties, we solve the thermal evolution equation and calculate the thermal
relaxation time for ample range of neutron star masses. By performing a
comprehensive study of neutron stars just above the onset of the direct Urca
process we show that stars under these conditions exhibit a peculiar thermal
relaxation behavior. We demonstrate that such stars exhibit an abnormally late
relaxation time, characterized by a second drop of its surface temperature
taking place a later ages. We qualify such behavior by showing that it is
associated with limited spatial distribution of the DU process is such stars.
We show that as the star's mass increase, the DU region also grows and the
start exhibits the expected behavior of fast cooling stars. Finally we show
that one can expect high relaxation times for stars in which the DU process
takes place in a radius not larger than 3 km.
