  We study the effective action of hydrostatic response to torsion in the
absence of spin connections in gapped $\left(2+1\right)$-dimensional
topological phases. In previous studies, a torsional Chern-Simons term with a
temperature-squared ($T^2$) coefficient was proposed as an alternative action
to describe thermal Hall effect with the idea of balancing the diffusion of
heat by a torsional field. However, the question remains whether this action
leads to local bulk thermal response which is not suppressed by the gap. In our
hydrostatic effective action, we show that the $T^2$ bulk term is invariant
under variations up to boundary terms considering the back reaction of the
geometry on local temperature, which precisely describes the edge thermal
current. Furthermore, there is no boundary diffeomorphism anomalies and bulk
inflow thermal currents at equilibrium and therefore no edge-to-edge adiabatic
thermal current pumping. These results are in consistent with exponentially
suppressed thermal current for gapped phases.
