  Enabling on-demand control of heat flow is key for the development of
next-generation electronic devices, solid-state heat pumps, and thermal logic.
However, precise and agile tuning of the relevant microscopic material
parameters for adjusting thermal conductivities remains elusive. Here, we study
several single crystals of lanthanum aluminate (LaAlO$_{3}$) with different
domain structures and show that ferroelastic domain walls behave as boundaries
that act like efficient controllers to govern thermal conductivity. At low
temperature (3 K), we demonstrate a fivefold reduction in thermal conductivity
induced by domain walls orthogonal to the heat flow and a twofold reduction
when they are parallel to the heat flow. Atomistic calculations fully support
this experimental observation. By breaking down phonon scattering mechanisms,
we also analyze the temperature dependence of the thermal conductivity to
derive a quantitative relation between thermal conductivity variations and
domain wall organization and density.
