  In this work, we study the effects of random temperature fluctuations on the
equation of state of a non-interacting, relativistic fermion gas by means of
the replica method. This picture provides a conceptual model for a
non-equilibrium system, depicted as an ensemble of subsystems at different
temperatures, randomly distributed with respect to a given mean value. We then
assume the temperature displays stochastic fluctuations $T = T_0 + \delta T$
with respect to its ensemble average value $T_0$, with zero mean
$\overline{\delta T} = 0$ and standard deviation $\overline{\delta T^2} =
\Delta$. By means of the replica method, we obtain the average grand canonical
potential, leading to the equation of state of the fermion gas expressed in
terms of the excess pressure caused by these fluctuations with respect to the
ideal gas at uniform temperature. We further extend our results for the ideal
Bose gas as well. Our findings reveal an increase in pressure as the system's
ensemble average temperature $T_0$ rises, consistently exceeding the pressure
observed in an equilibrium state. Finally, we explore the implications for the
deconfinement transition in the context of the simple Bag model, where we show
that the critical temperature decreases.
