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No interaction between DM and baryonic matter is assumed, except the gravitational one. Using the two-fluid formalism, we show that DM accumulated in the core of a star pulls inwards the outer baryonic layers of the star, increasing the baryonic density in the NS core. As a result, it significantly affects the star\u2019s thermal evolution by triggering an early onset of the direct Urca (DU) process and modifying the photon emission from the surface caused by the decrease of the radius. Thus, due to the gravitational pull of DM, the DU process becomes kinematically allowed for stars with lower masses. Based on these results, we discuss the importance of NS observations at different distances from the Galactic Centre. 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