  In Ref. [S. Jalalzadeh, Phys. Lett. B 829 (2022) 137058], Jalalzadeh
established that the thermodynamical entropy of a quantum-deformed black hole
with horizon area $A$ can be written as $S_q=\pi\sin\left(\frac{A}{8G\mathcal
N} \right)/\sin\left(\frac{\pi}{2\mathcal N} \right)$, where $\mathcal
N=L_q^2/L_\text{P}^2$, $L_\text{P}$ being the Planck length and $L_q$ denoting,
generically, the q-deformed cosmic event horizon distance $L_q$. Motivated by
this, we now extend the framework constructed in [S. Jalalzadeh, Phys. Lett. B
829 (2022) 137058] towards the Friedmann and Raychaudhuri equations describing
spatially homogeneous and isotropic universe dynamics. Our procedure in this
paper involves a twofold assumption. On the one hand, we take the entropy
associated with the apparent horizon of the Robertson-Walker universe in the
form of the aforementioned expression. On the other hand, we assume that the
unified first law of thermodynamics, $dE=TdS+WdV$, holds on the apparent
horizon. Subsequently, we find a novel modified cosmological scenario
characterized by quantum-deformed (q-deformed) Friedmann and Raychaudhuri
equations containing additional components that generate an effective dark
energy sector. Our results indicate an effective dark energy component, which
can explain the Universe's late-time acceleration. Moreover, the Universe
follows the standard thermal history, with a transition redshift from
deceleration to acceleration at $z_\text{tran}=0.5$. More precisely, according
to our model, at a redshift of $z = 0.377$, the effective dark energy dominates
with a de Sitter universe in the long run. We include the evolution of
luminosity distance, $\mu$, the Hubble parameter, $H(z)$, and the deceleration
parameter, $q(z)$, versus redshift. Finally, we have conducted a comparative
analysis of our proposed model with others involving non-extensive entropies.
