  Recently, Barrow accounts for the quantum gravitational effects to the black
hole surface. Thus the conventional area-entropy relation has modified,
$S=(A/A_{0})^{1+\Delta/2},$ with an exponent $\Delta$, ranges $0\le\Delta\le1$,
quantifies the amount of quantum gravitational deformation effect to the black
hole surface. In recent literature, this horizon entropy has been extended to
the cosmological context. Following this, we consider an n+1 dimensional
non-flat universe with an apparent horizon as the boundary with appropriate
temperature and associated entropy is Barrow entropy. We derived the modified
form of the law of emergence from the equilibrium and non-equilibrium
thermodynamic principles. Later studied the entropy maximization condition due
to the modified law of emergence. On distinguishing the obtained result, it
speculates that in order to hold the energy-momentum conservation, the universe
with Barrow entropy as the horizon entropy should have non-equilibrium
behaviour with an additional entropy production. However, the additional
entropy production rate decreases over time, so the system eventually
approaches equilibrium. Because of this, the constraint relation for entropy
maximization looks similar for both equilibrium and non-equilibrium approaches.
