Planckian corrections to the Friedmann flat equations from thermodynamics at the apparent horizon

In this paper we use our recently generalized black hole entropy formula to propose a quantum version of the Friedmann equations. In particular, starting from the differential version of the first law of thermodynamics, we are able to find planckian (non commutative) corrections to the Friedmann flat equations. The so modified equations are formally similar to the ones present in Gauss-Bonnet gravity, but in the ordinary 3+1 dimensions. As a consequence of these corrections, by considering negative fluctuations in the internal energy that are allowed by quantum field theory, our equations imply a maximum value both for the energy density $\rho$ and for the Hubble flow $H$, i.e. the big bang is ruled out. Conversely, by considering positive quantum fluctuations, we found no maximum for $\rho$ and $H$. Nevertheless, by starting with an early time energy density $\rho\sim 1/t^2$, we obtain a value for the scale factor $a(t)\sim e^{\sqrt{t}}$, implying a finite planckian universe at $t=0$, i.e. the point-like big bang singularity is substituted by a universe of planckian size at $t=0$. Finally, we found possible higher order planckian terms to our equations together with the related corrections of our generalized Bekenstein-Hawking entropy.