The interior of a neutron star is expected to be occupied by a neutron $^3P_2$ superfluid, which is the condensate of spin-triplet $p$-wave Cooper pairs of neutrons with total angular momentum $J=2$. Here we investigate the thermodynamic stability of $^3P_2$ superfluids in a neutron-star interior under a strong magnetic field. Using the theory incorporating the finite size correction of neutron Fermi surface, we show that the spin-polarized phases of $^3P_2$ superfluids, the magnetized biaxial nematic phase and the ferromagnetic phase, appear in high temperatures and high magnetic fields. These phases were missed in the previous studies using the quasiclassical approximation in which dispersions of neutrons are linearized around the Fermi surface. In particular, the ferromagnetic phase, which is the condensation of Cooper-paired neutrons with fully polarized spins, appears between the normal phase and the biaxial nematic phase and enlarge the thermodynamic stability of $^3P_2$ superfluids under strong magnetic fields. Furthermore, we present the augmented Ginzburg-Landau theory that incorporates the thermodynamic stability of spin-polarized $^3P_2$ superfluid phases.
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