Due to the diverging density of states, flat-band superconductors, such as twisted bilayer graphene, are promising candidates for achieving high superconducting transition temperatures. Understanding their electrical and thermal properties is therefore of great importance. While the superfluid weight is determined by the quantum geometry in crystal-momentum space, we show that the thermal Meissner stiffness can be expressed in terms of the quantum metric in the space of an external gravitomagnetic vector potential, which is called the heat quantum metric. Physically, the heat quantum metric characterizes fluctuations of the heat polarization and defines a characteristic energy scale in quantum materials. Because of the positive semidefiniteness of the quantum metric, we further identify a Wiedemann-Franz-type inequality for the ratio of the superfluid weight to the thermal Meissner stiffness. Finally, we highlight the implications for the experimental identification of pairing symmetries based on low-temperature scaling laws of the superfluid weight and other observables.