  Taking into account the zero-point length correction to the gravitational
potential and using the entropic force scenario, we derive a modified
expression for entropy that includes zero-point length correction terms. We
then apply the thermodynamics-gravity conjecture and the emergent gravity
scenario to derive the modified Friedmann equations. These equations describe
the evolution of the universe with zero-point length corrections included in
the entropy expression. Further, we examine the time evolution of the total
entropy, including the entropy of the apparent horizon and the matter field
entropy inside the horizon and find that the generalized second law of
thermodynamics is satisfied. Besides, we investigate cosmological implications
of the modified cosmology through zero-point length. We study several cases,
including radiation dominated era, matter dominated era and universe filled
with matter and cosmological constant. We observe that the zero-point length
correction does not change the general profile of the universe evolution,
however, it shifts the time of the phase transition in a universe filled with
matter and cosmological constant. We explore the age of the universe for our
model and observe that the predicted age of the universe becomes larger
compared to the standard cosmology. By calculating the explicit form of Ricci
and Kretchmann invariants, we confirm that in our model, the initial
singularity of the universe is removed. This is an expected result, because the
main motivation for considering zero-point length correction in the
gravitational potential is to remove singularity at the origin.
