  Many discussions in the literature of spacetimes with more than one Killing
horizon note that some horizons have positive and some have negative surface
gravities, but assign to all a positive temperature. However, the first law of
thermodynamics then takes a non-standard form. We show that if one regards the
Christodoulou and Ruffini formula for the total energy or enthalpy as defining
the Gibbs surface, then the rules of Gibbsian thermodynamics imply that
negative temperatures arise inevitably on inner horizons, as does the
conventional form of the first law. We provide many new examples of this
phenomenon, including black holes in STU supergravity. We also give a
discussion of left and right temperatures and entropies, and show that both the
left and right temperatures are non-negative. The left-hand sector contributes
exactly half the total energy of the system, and the right-hand sector
contributes the other half. Both the sectors satisfy conventional first laws
and Smarr formulae. For spacetimes with a positive cosmological constant, the
cosmological horizon is naturally assigned a negative Gibbsian temperature. We
also explore entropy-product formulae and a novel entropy-inversion formula,
and we use them to test whether the entropy is a super-additive function of the
extensive variables. We find that super-additivity is typically satisfied, but
we find a counterexample for dyonic Kaluza-Klein black holes.
