  We study a stationary and axisymmetric binary system composed of two
identical Kerr black holes, whose physical parameters satisfy the Smarr
thermodynamic formula. Then, we use the formalism of geometrothermodynamics to
show that the spatial distance between the black holes must be considered as a
thermodynamic variable. We investigate the main thermodynamic properties of the
system by using the contact structure of the phase space, which generates the
first law of thermodynamics and the equilibrium conditions. The phase
transition structure of the system is investigated through the curvature
singularities of the equilibrium space. It is shown that the thermodynamic and
stability properties and the phase transition structure of the binary system
strongly depend on the distance between the black holes.
