  We report the in-plane and out-of-plane heat conductivity of the
antiferromagnetic spin-orbit induced Mott insulator Sr$_2$IrO$_4$ with
$J_\mathrm{eff} = 1/2$. Our data reveal clear-cut evidence for magnetic heat
transport within the IrO$_2$ planes which provides the unique possibility to
analyze the thermal occupation and scattering of $J_\mathrm{eff} = 1/2$
pseudospin excitations. The analysis of the magnetic heat conductivity yields a
low-temperature ($T \lesssim 75$ K) magnetic mean free path $l_\mathrm{mag}
\sim 32$ nm, consistent with boundary scattering. Upon heating towards room
temperature, the mean free path strongly decreases by one order of magnitude
due to thermally activated scattering of the pseudospin excitations. The latter
reveals that the coupling of these excitations to the lattice is radically
different from that of $S = 1/2$-excitations in cuprate analogs.
