  Building on the idea of Tolman and Ehrenfest that heat has weight, Luttinger
established a deep connection between gravitational fields and thermal
transport. However, this relation does not include anomalous quantum
fluctuations that become paramount in strongly curved spacetime. In this work,
we revisit the celebrated Tolman-Ehrenfest and Luttinger relations and show how
to incorporate the quantum energy scales associated with these fluctuations,
captured by gravitational anomalies of quantum field theories. We point out
that such anomalous fluctuations naturally occur in the quantum atmosphere of a
black hole. Our results reveal that analogous fluctuations are also observable
in thermal conductors in flat-space time provided local temperature varies
strongly. As a consequence, we establish that the gravitational anomalies
manifest themselves naturally in non-linear thermal response of a quantum wire.
In addition, we propose a systematic way to identify thermal analogues of black
hole's anomalous quantum fluctuations associated to gravitational anomalies. We
identify their signatures in propagating energy waves following a thermal
quench, as well as in the energy density of heating Floquet states induced by
repeated quenches.
