  We study the thermodynamics and critical behavior of neutron $^3P_2$
superfluids in the inner cores of neutron stars. $^3P_2$ superfluids offer a
rich phase diagram including uniaxial/biaxial nematic phases, the ferromagnetic
phase, and the cyclic phase. Using the Bogoliubov-de Gennes (BdG) equation as
superfluid Fermi liquid theory, we show that a strong (weak) magnetic field
drives the first (second) order transition from the dihedral-two biaxial
nematic phase to dihedral-four biaxial nematic phase in low (high)
temperatures, and their phase boundaries are divided by the critical endpoint
(CEP). We demonstrate that the set of critical exponents at the CEP satisfies
the Rushbrooke, Griffiths, and Widom equalities, indicating a new universality
class. At the CEP, the $^3P_2$ superfluid exhibits critical behavior with
nontrivial critical exponents, indicating a new universality class.
Furthermore, we find that the Ginzburg-Landau (GL) equation up to the 8th-order
expansion satisfies three equalities and properly captures the physics of the
CEP. This implies that the GL theory can provide a tractable way for
understanding critical phenomena which may be realized in the dense core of
realistic magnetars.
