Entropy of non-extremal STU black holes: the F-invariant unveiled

We find a manifestly U-duality invariant formula for the Bekenstein-Hawking entropy of the most general 4 dimensional, stationary, asymptoticaly flat, non-extremal STU black holes constructed recently by Chow and Comp`ere. The expression is entirely in terms of asymptotic charges. It involves the “scalar charges” of the black hole which still need to be solved in terms of the dyonic charges and the mass. We discuss how the formula reduces to some of the known results as the Klauza-Klein black hole and the dilute gas limit of Cveti\v{c} and Larsen. We give the expected generalization to an $E_{7(7)}$ invariant in the case of maximal $\mathcal{N}=8$ supergravity.