  The Heisenberg-Kitaev (HK) model on various lattices have attracted a lot of
attention because they may lead to exotic states such as quantum spin liquid
and topological orders. The rare-earth-based kagome lattice (KL) compounds
$\rm{Mg_2RE_3Sb_3O_{14}}$ $\rm{(RE=Gd, Er)}$ and $\rm{(RE=Nd)}$ have ${\bf
q=0}$, $120^\circ$ order and canted ferromagnetic (CFM) order, respectively.
Interestingly, the HK model on the KL has the same ground state long-range
orders. In the theoretical phase diagram, the CFM phase resides in a continuous
parameter region and there is no phase change across special parameter points,
such as the Kitaev ferromagnetic (KFM) point, the ferromagnetic (FM) point and
its dual FM point. However, a ground state property cannot distinguish a system
with or without topological nontrivial excitations and related phase
transitions. Here, we study the topological magnon excitations and related
thermal Hall conductivity in the HK model on the KL with CFM order. The CFM
phase can be divided into two regions related by the Klein duality, with the
self dual KFM point as their boundary. We find that the scalar spin chirality
which is intrinsic in the CFM order changes sign across the KFM point. This
leads to the opposite Chern numbers of corresponding magnon bands in the two
regions, and also the sign change of the magnon thermal Hall conductivity.
