  Considering Verlinde's entropic gravity proposal, we focus the effects of
fermionic $q$ deformation on the Einstein's field equations and Friedmann
equations. For this purpose, we represent the thermodynamical properties of the
deformed fermion gas model in two-dimensional space. To describe the behavior
of the quantum black holes, deformed Einstein field equations are derived with
the help of deformed entropy function. Moreover, deformed Friedmann equations
are investigated by using Friedmann-Robertson-Walker (FRW) metric. We also
present the effective ones of the energy density, the pressure, and the
equation of state for the dark energy. Lastly, we derive an analytical
expression of the effective density parameter of dark energy.
