  We explore spin dynamics in Cu(1,3-bdc), a quasi-2D topological magnon
insulator. The results show that the thermal evolution of Land\'e $g$-factor
($g$) is anisotropic: $g_\textrm{in-plane}$ reduces while
$g_\textrm{out-plane}$ increases with increasing temperature $T$. Moreover, the
anisotropy of the $g$-factor ($\Delta g$) and the anisotropy of saturation
magnetization ($\Delta M_\textrm{s}$) are correlated below 4 K, but they
diverge above 4 K. We show that the electronic orbital moment contributes to
the $g$ anisotropy at lower $T$, while the topological orbital moment induced
by thermally excited spin chirality dictates the $g$ anisotropy at higher $T$.
Our work suggests an interplay among topology, spin chirality, and orbital
magnetism in Cu(1,3-bdc).
