  This paper investigates generalized thermodynamic relationships in physical
systems where relevant macroscopic variables are determined by the exponential
Kolmogorov-Nagumo average. We show that while the thermodynamic entropy of such
systems is naturally described by R\'{e}nyi's entropy with parameter $\gamma$,
an ordinary Boltzmann distribution still describes their statistics under
equilibrium thermodynamics. Our results show that systems described by
exponential Kolmogorov-Nagumo averages can be interpreted as systems originally
in thermal equilibrium with a heat reservoir with inverse temperature $\beta$
that are suddenly quenched to another heat reservoir with inverse temperature
$\beta' = (1-\gamma)\beta$. Furthermore, we show the connection with
multifractal thermodynamics. For the non-equilibrium case, we show that the
dynamics of systems described by exponential Kolmogorov-Nagumo averages still
observe a second law of thermodynamics and the H-theorem. We further discuss
the applications of stochastic thermodynamics in those systems -- namely, the
validity of fluctuation theorems -- and the connection with thermodynamic
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