By coupling Bardeen-Cooper-Schrieffer (BCS) theory with isolated bands to an external gravitomagnetic vector potential via a gravitomagnetic Peierls substitution, we identify a quantum-geometric contribution to the electronic contribution of the thermal conductivity. This contribution is governed by the quantum metric in the parameter space spanned by the components of the external gravitomagnetic vector potential which corresponds to a weighted quantum metric in momentum space. In the flat-band limit, we establish an upper and lower Wiedemann-Franz-type bound for the ratio of thermal Meissner stiffness and electric Meissner stiffness (superfluid weight), whose prefactors are provided by the extrema of the squared energy offsets of the outer single-particle bands of the system. Similarly to the superfluid weight, this also leads to a lower bound of the thermal Meissner stiffness in terms of the Chern number. Our results apply to both superconductors and other fermionic superfluids.

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• Multiband superconductivity
• Quantum geometry
• Superfluid density
• Thermal conductivity
• Superconductors
• BCS theory
• Mean field theory