Finite heat reservoir capacity and temperature fluctuations lead to modification of the well known canonical exponential weight factor. Requiring that the corrections least depend on the one-particle energy, we derive a deformed entropy, K(S). The resulting formula contains the Boltzmann-Gibbs, the Renyi and the Tsallis formulas as particular cases. For extreme large fluctuations (compared to the Gaussian case) a new, parameter-free entropy - probability relation emerges. This formula and the corresponding canonical equilibrium distribution are nearly Boltzmannian for high probability, but deviate from the classical result for low probability. In the extreme large fluctuation limit the canonical distribution resembles for low probability the cumulative Gompertz distribution.
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