Thermodynamic Derivation of the Tsallis and R\'enyi Entropy Formulas and the Temperature of Quark-Gluon Plasma

We derive Tsallis entropy, Sq, from universal thermostat independence and obtain the functional form of the corresponding generalized entropy-probability relation. Our result for finite thermostats interprets thermodynamically the subsystem temperature, T1, and the index q in terms of the temperature, T, entropy, S, and heat capacity, C of the reservoir as T1 = T exp(-S/C) and q = 1 - 1/C. In the infinite C limit, irrespective to the value of S, the Boltzmann-Gibbs approach is fully recovered. We apply this framework for the experimental determination of the original temperature of a finite thermostat, T, from the analysis of hadron spectra produced in high energy collisions, by analyzing frequently considered simple models of the quark-gluon plasma.