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We study different scenarios of topological phase transitions in the vicinityof the \pi^0 condensation point in neutron matter. The transitions occurbetween the Fermi-liquid state and a topologically different one with twosheets of the Fermi surface. Two possibilities of a rearrangement ofquasiparticle degrees of freedom are shown: the first-order topological phasetransition and the second-order one. The order of the phase transition is foundto 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 thatthe topology of the quasiparticle momentum distribution is mainly determined bythe neutron matter density, while the temperature T is essential in a narrowdensity region. A simple explanation of the first-order topological phasetransition at T=0 is given.

Different scenarios of topological phase transitions in homogeneous neutron matter