We discuss the thermalization process of the neutron stars crust described by solving the heat transport equation with a microscopic input for the specific heat of baryonic matter. The heat equation is solved with initial conditions specific to a rapid cooling of the core. To calculate the specific heat of inner crust baryonic matter, i.e., nuclear clusters and unbound neutrons, we use the quasiparticle spectrum provided by the Hartree-Fock-Bogoliubov approach at finite temperature. In this framework we analyze the dependence of the crust thermalization on pairing properties and on cluster structure of inner crust matter. It is shown that the pairing correlations reduce the crust thermalization time by a large fraction. The calculations show also that the nuclear clusters have a non-negligible influence on the time evolution of the surface temperature of the neutron star.