$E_{7(7)}$ invariant non-extremal entropy

The entropy of generic non-extremal dyonic black holes in the STU model has been shown to admit a remarkably universal form. The missing invariant in the formula was recently identified by S\'arosi using the formalism of quantum entanglement as well as a higher dimensional embedding of the U-duality group. Here, we express the non-extremal black hole entropy in the STU model in terms of U-duality covariant tensors. We then provide the extension to the most general non-extremal black hole of ungauged $\mathcal N= 8$ supergravity using $E_{7(7)}$ invariants. We also conjecture a generalization for ungauged $\mathcal N = 2$ supergravity coupled to vector multiplets with arbitrary cubic prepotential. The most general rotating dyonic black hole solution of the STU model with all scalar moduli turned on is provided in an appendix.