  We report studies of thermal conductivity as functions of magnetic field and
temperature in the helimagnetic insulator Cu$_2$OSeO$_3$ that reveal novel
features of the spin-phase transitions as probed by magnon heat conduction. The
tilted conical spiral and low-temperature skyrmion phases, recently identified
in small-angle neutron scattering studies, are clearly identified by sharp
signatures in the magnon thermal conductivity. Magnon scattering associated
with the presence of domain boundaries in the tilted conical phase and regions
of skyrmion and conical-phase coexistence are identified.
