  We study the thermal transport in two-dimensional systems with a nontrivial
Berry curvature texture. The physical realizations are many: for a sake of
definiteness we consider undoped graphene gapped by the presence of an aligned
hexagonal-Boron-Nitride substrate. The same phenomenology applies, i.e., to
surface states of 3D topological insulators in the presence of a uniform
magnetization. We find that chiral valley-polarized second-sound collective
modes propagate along the edges of the system. The localization length of the
edge modes has topological origin stemming from the anomalous velocity term in
the quasiparticle current. At low temperature, the single-particle contribution
to the transverse thermal conductance is exponentially suppressed and only
second-sound modes carry heat along the boundary. A sharp change in the
behavior of the thermal Hall conductance, extracted from nonlocal measurements
of the temperature along the edge, marks the onset of ballistic heat transport
due to second-sound edge modes.
