  We study different scenarios of topological phase transitions in the vicinity
of the \pi^0 condensation point in neutron matter. The transitions occur
between the Fermi-liquid state and a topologically different one with two
sheets of the Fermi surface. Two possibilities of a rearrangement of
quasiparticle degrees of freedom are shown: the first-order topological phase
transition and the second-order one. The order of the phase transition is found
to be strongly dependent on the value of the critical wave vector of the soft
\pi^0 mode. The thermodynamics of the system is also studied. It is shown that
the topology of the quasiparticle momentum distribution is mainly determined by
the neutron matter density, while the temperature T is essential in a narrow
density region. A simple explanation of the first-order topological phase
transition at T=0 is given.
