Temperature and free energy of multi-black-hole systems

In the present paper, we compute the Euclidean action of a generic system consisting of N arbitrary Kerr-Newman black holes located on the symmetry axis and separated from each other by massless struts. This allows us to introduce for the first time the notion of "Hawking average temperature" T ^ of the multi-black-hole system via the condition of vanishing the entire set of terms involving the singular horizons due to periodic time, and the resulting formula for this temperature contains solely the surface gravities κ_i and horizon areas A_i^H of the black hole constituents. We also show that the corresponding expression for the free energy of the system defined by T ^ is consistent with the first law of thermodynamics and Smarr mass relations.