  Motivated by a previous "$sd^2$-graphene" study, the pairing symmetry in the
superconducting state and the thermal Hall conductivity are investigated by a
self-consistent Bogoliubov--de Gennes approach on the kagome lattice with
intrinsic spin-orbit coupling near van Hove fillings. While the topologically
trivial state with broken time-reversal symmetry appears in the absence of
spin-orbit coupling, the highest flat band becomes dispersive with a hexagonal
symmetry due to spin-orbit coupling, which leads to a topological
superconducting state. Since the thermal Hall conductivity in the
low-temperature limit is associated with the topological property of
time-reversal symmetry breaking superconductors, we study its temperature
dependence near van Hove fillings. In particular, the pairing symmetry in the
highest flat band is sensitive to the amplitudes of spin-orbit coupling and the
attractive interaction, which is reflected remarkably in the thermal Hall
conductivity. The obtained result may enable us to investigate the stable
superconducting state on the kagome lattice.
