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The response of solids to temperature gradients is often described in termsof a gravitational analogue: the effect of a space-dependent temperature ismodeled using a space dependent metric. We investigate the validity of thisapproach in describing the bulk response of quantum Hall states and othergapped chiral topological states. To this end, we consider the prototypicalHaldane model in two different cases of (i) a space-dependent electrostaticpotential and gravitational potential and (ii) a space-dependent temperatureand chemical potential imprinted by a weak coupling to non-interacting electronbaths and phonons. We find that the thermal analogue is \textit{invalid}; whilea space dependent gravitational potential induces transverse energy currentsproportional to the third derivative of the gravitational potential, theresponse to an analogous temperature profile vanishes in limit of weak couplingto the thermal bath. Similarly, the Einstein relation, the analogy between theelectrostatic potential and the internal chemical potential, is not valid insuch a setup.

Thermal Hall response: violation of gravitational analogues and Einstein relations