  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 the {\it Hawking average temperature} (HAT) $\hat T$ 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 $\kappa_i$ and
horizon areas $A^H_i$ of the black hole constituents. We also show that the
corresponding expression for the {\it free energy} of the system defined by
$\hat T$ is consistent with the first law of thermodynamics and Smarr mass
relations.
