  The article addresses a possibility of obtaining cosmologically relevant
effects from the quantum nature of the Hubble horizon. Following the
observation made by E.~Bianchi and C.~Rovelli in Phys.\ Rev.\ D {\bf 84} (2011)
027502 we explore relationship between the Planck scale discreteness of the
Hubble horizon and deformations of the symmetry of rotations. We show that the
so-called $q$-deformations in a natural way lead to a mechanism of condensation
in the very early Universe. We argue that this provides a possible resolution
of the problem of initial homogeneity at the onset of inflation. Furthermore,
we perform entropic analysis of the quantum Hubble horizon and show that the
$\Lambda$CDM model may arise from linearly (in area of the horizon) corrected
Bekenstein-Hawking entropy. Based on this, we have shown that the current
accelerating expansion can be associated with the entropy decrease in the
Hubble volume. The presented results open new ways to explore relation between
the Planck scale effects and observationally relevant features of our Universe.
