  The response of solids to temperature gradients is often described in terms
of a gravitational analogue: the effect of a space-dependent temperature is
modeled using a space dependent metric. We investigate the validity of this
approach in describing the bulk response of quantum Hall states and other
gapped chiral topological states. To this end, we consider the prototypical
Haldane model in two different cases of (i) a space-dependent electrostatic
potential and gravitational potential and (ii) a space-dependent temperature
and chemical potential imprinted by a weak coupling to non-interacting electron
baths and phonons. We find that the thermal analogue is \textit{invalid}; while
a space dependent gravitational potential induces transverse energy currents
proportional to the third derivative of the gravitational potential, the
response to an analogous temperature profile vanishes in limit of weak coupling
to the thermal bath. Similarly, the Einstein relation, the analogy between the
electrostatic potential and the internal chemical potential, is not valid in
such a setup.
