Consistency of the structure of Legendre transform in thermodynamics with the Kolmogorov–Nagumo average

• The robustness of the Legendre structure has been shown within the KN average.
• The relationships between the expectation values and the Lagrange multipliers still hold in the present formalism.
• The universality of the Gibbs equation and other thermodynamic relations are unaffected by the structure of the average used.

We show the robustness of the structure of Legendre transform in thermodynamics against the replacement of the standard linear average with the Kolmogorov–Nagumo nonlinear average to evaluate the expectation values of the macroscopic physical observables. The consequence of this statement is twofold: 1) the relationships between the expectation values and the corresponding Lagrange multipliers still hold in the present formalism; 2) the universality of the Gibbs equation as well as other thermodynamic relations are unaffected by the structure of the average used in the theory.