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

Anomalous Luttinger equivalence between temperature and curved spacetime: From black hole's atmosphere to thermal quenches