  According to the quantum deformation approach to quantum gravity, the
thermodynamical entropy of a quantum-deformed (q-deformed) black hole with
horizon area $A$ established by Jalalzadeh is expressed as $S_q = \pi\sin
\left( \frac{A}{8G\mathcal N} \right) /\sin\left(\frac{\pi}{2\mathcal
N}\right)$, where $\mathcal N=L_q^2/L_{p}^2$ is the q-deformation parameter,
$L_{p}$ denotes the Planck length, and $L_q$ denotes the quantum-deformed
cosmic apparent horizon distance. In this paper, assuming that the q-deformed
entropy is associated with the apparent horizon of the
Friedmann-Robertson-Walker (FRW) universe, we derive the Friedmann equation
from the unified first law of thermodynamics, ${dE = T_q dS_q + WdV}$. And this
one obtained is in line with the Friedmann equation derived from the law of
emergence proposed by Padmanabhan. It clearly shows the connection between the
law of emergence and the unified first law of thermodynamics. Subsequently, in
the context of the q-deformed horizon entropy, we investigate the constraints
of entropy maximization, and result demonstrates the consistency of the law of
emergence with the maximization of the q-deformed horizon entropy. Hence, the
law of emergence can be understood as a tendency to maximize the q-deformed
horizon entropy.
