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We show the robustness of the structure of Legendre transform inthermodynamics against the replacement of the standard linear average with theKolmogorov-Nagumo nonlinear average to evaluate the expectation values of themacroscopic physical observables. The consequence of this statement is twofold:1) the relationships between the expectation values and the correspondingLagrange multipliers still hold in the present formalism; 2) the universalityof the Gibbs equation as well as other thermodynamic relations are unaffectedby the structure of the average used in the theory.

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