Thermodynamics of two aligned Kerr-Newman black holes

Using the notion of thermodynamic length, the first law of thermodynamics is consistently derived for two binary configurations of equal Kerr-Newman black holes separated by a massless strut. Like in the electrostatic systems of two Reissner-Nordstr"om black holes and stationary vacuum systems of two Kerr black holes considered earlier, the thermodynamic length $\mathrm{\ell }$ turns out to be defined by the same simple formula $\mathrm{\ell }=L\exp ({\gamma }_0)$, $L$ being the coordinate length of the strut and ${\gamma }_0$ the value of the metric function $\gamma$ on the strut, which permits the elaboration of $\mathrm{\ell }$ in a concise analytic form. The expression of the free energy in the case of two generic Kerr-Newman black holes is also proposed.