  As rotating neutron stars slow down, the pressure and the density in the core
region increase due to the decreasing centrifugal forces and phase transitions
may occur in the center. We extract the analytic behavior near the critical
angular velocity $\Omega_0$, where the phase transitions occur in the center of
a neutron star, and calculate the moment of inertia, angular velocity, rate of
slow down, braking index, etc. For a first order phase transition these
quantities have a characteristic behavior, e.g., the braking index diverges as
$\sim(\Omega_0-\Omega)^{-1/2}$. Observational consequences for first, second
and other phase transitions are discussed.
