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We consider stationary, non-extremal black holes in 11-dimensionalsupergravity having isometry group $\mathbb{R} \times U(1)^8$. We prove thatsuch a black hole is uniquely specified by its angular momenta, its electriccharges associated with the various 7-cycles in the manifold, together withcertain moduli and vector valued winding numbers characterizing the topologicalnature of the spacetime and group action. We furthermore establish interesting,non-trivial, relations between the thermodynamic quantities associated with theblack hole. These relations are shown to be a consequence of the hidden$E_{8(+8)}$ symmetry in this sector of the solution space, and are distinctfrom the usual "Smarr-type" formulas that can be derived from the first law ofblack hole mechanics. We also derive the "physical process" version of thisfirst law applicable to a general stationary black hole spacetime without anysymmetry assumptions other than stationarity, allowing in particular arbitraryhorizon topologies. The work terms in the first law exhibit the topology of thehorizon via the intersection numbers between cycles of various dimensions.

Black hole uniqueness theorems and new thermodynamic identities in eleven dimensional supergravity