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We probe the low-temperature behavior of a system of quantum bouncers as atheoretical model for ultracold neutrons within a low energy modified versionof the standard quantum mechanics, due to the gravitational effects. Working inone dimension, the energy spectrum and bound states of a deformed quantumbouncer are obtained using the first-order WKB approximation, granted the verylow energy regime of the particle. In this manner, we can study energy levelsof a system of ultracold neutrons as an informative probe towards exploring thelow energy manifestation of semi-classical quantum gravitational effects. Ourcalculated energy levels of ultracold neutrons are in accordance with theobserved energy levels, as obtained in the famous Nesvizhevsky \emph{et al.}experiment, with a negative constant deformation, as dependent on thedeformation parameter. In advance, we tackle modified thermodynamics of asystem of quantum bouncers in the infrared regime via an ensemble theory bothin one dimension and also three dimensions, to seek for any trace of aneffective, thermodynamic dimensional reduction in this low energy regime ofsemi-classical quantum gravity. While the issue of dimensional reduction hasbeen essentially assigned to the high energy regime, here we show that there isa trace of an effective, thermodynamic dimensional reduction in infrared regimewith one important difference: in the high energy regime, the dimensionalreduction effectively occurs from $D=3$ to $D=1$, but here, in this low energyregime, there is a trace of thermodynamic dimensional reduction from $D=3$ to$D=2$.

IR-deformed thermodynamics of quantum bouncers and the issue of dimensional reduction