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The Urca cooling strength of $^{63}$Fe-$^{63}$Mn exhibits a large range of variation due to the the uncertainty in the spin-parity of ground state of $^{63}$Fe, which is approaching the Island of Inversion at $N=40$, characterized by large nuclear deformation. To investigate the cooling effect on the thermal evolution of neutron star crust, we preformed simulations by using models for computing neutron star structure and evolution. Cooling strength obtained from various experiments on the $\beta ^{-}$ decay of $^{63}$Mn are included in this work. When adopting data recommended by NNDC, the cooling strength of $^{63}$Fe-$^{63}$Mn pair could have significant impact on the crust thermal evolution of neutron star including the crust cooling in quiescent phase and on the superburst ignition. More precise measurements of the $\beta^{-}$ decay of $^{63}$Mn are needed to better understand the Urca cooling effect of the $^{63}$Mn-$^{63}$Fe pair, as well as other potential Urca pairs in the neutron-rich region.