We discuss the thermalization process of the neutron star's 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, that is, 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