  We have theoretically investigated transport properties of the classical
Heisenberg antiferromagnet on the triangular lattice in which a
binding-unbinding topological transition of $\mathbb{Z}_2$ vortices is
predicted to occur at a finite temperature $T_v$. It is shown by means of the
hybrid Monte-Carlo and spin-dynamics simulations that the longitudinal
spin-current conductivity exhibits a divergence at $T_v$, while the thermal
conductivity only shows a monotonic temperature dependence with no clear
anomaly at $T_v$. The significant enhancement of the spin-current conductivity
is found to be due to the rapid growth of the spin-current-relaxation time
toward $T_v$, which can be understood as a manifestation of the topological
nature of the free $\mathbb{Z}_2$ vortex whose lifetime gets longer toward
$T_v$. The result suggests that the spin-current measurement is a promising
probe to detect the $\mathbb{Z}_2$-vortex topological transition which has
remained elusive in experiments.
