  Globally neutral neutron stars, obtained from the solution of the called
Einstein-Maxwell-Thomas-Fermi equations that account for all the fundamental
interactions, have been recently introduced. These configurations have a more
general character than the ones obtained with the traditional
Tolman-Oppenheimer-Volkoff, which impose the condition of local charge
neutrality. The resulting configurations have a less massive and thinner crust,
leading to a new mass-radius relation. Signatures of this new structure of the
neutron star on the thermal evolution might be a potential test for this
theory. We compute the cooling curves by integrating numerically the energy
balance and transport equations in general relativity, for globally neutral
neutron stars with crusts of different masses and sizes, according to this
theory for different core-crust transition interfaces. We compare and contrast
our study with known results for local charge neutrality. We found a new
behavior for the relaxation time, depending upon the density at the base of the
crust, $\rho_{\rm crust}$. In particular, we find that the traditional increase
of the relaxation time with the crust thickness holds only for configurations
whose density of the base of the crust is greater than $\approx 5\times
10^{13}$ g cm$^{-3}$. The reason for this is that neutron star crusts with very
thin or absent inner crust have some neutrino emission process blocked which
keep the crust hotter for longer times. Therefore, accurate observations of the
thermal relaxation phase of neutron stars might give crucial information on the
core-crust transition which may aid us in probing the inner
composition/structure of these objects.
