Emergence of cosmic space and horizon thermodynamics in the context of the quantum-deformed entropy

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 $S_q=\pi \sin \left(\frac{A}{8G\mathcal{N}}\right)\mathrm{/}\sin \left(\frac{\pi }{2\mathcal{N}}\right)$, where $\mathcal{N}=L_q^2\mathrm{/}{L}_p^2$ 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, $dE=T_{q}d{S}_q+WdV$. 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.