  We investigate the impact of asymmetric fermionic dark matter (DM) on the
thermal evolution of neutron stars (NSs), considering a scenario where DM
interacts with baryonic matter (BM) through gravity. Employing the two-fluid
formalism, our analysis reveals that DM accrued within the NS core exerts an
inward gravitational pull on the outer layers composed of BM. This
gravitational interaction results in a noticeable increase in baryonic density
within the core of the NS. Consequently, it strongly affects the star's thermal
evolution by triggering an early onset of the direct Urca (DU) processes,
causing an enhanced neutrino emission and rapid star cooling. Moreover, the
photon emission from the star's surface is modified due to a reduction of
radius. We demonstrate the effect of DM gravitational pull on nucleonic and
hyperonic DU processes that become kinematically allowed even for NSs of low
mass. We then discuss the significance of observing NSs at various distances
from the Galactic center. Given that the DM distribution peaks toward the
Galactic center, NSs within this central region are expected to harbor higher
fractions of DM, potentially leading to distinct cooling behaviors.
