We provide a thermodynamic meaning of the index of the γ-entropy, which is a non-additive information measure devised in 1970s and is less known to physics. It is intimately related to the free energy as was previously shown for the Rényi entropy, which is extensive (additive), where the entropic index has an interpretation of the ratio of different temperatures. We conclude that this picture holds true for other generalized entropies irrespective of their nonextensivity.