Thermodynamic behavior of cosmological models with fractional entropy

We investigate the thermodynamic and phenomenological implications of a cosmological model governed by fractional entropy applied to the apparent horizon of a flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) universe. By utilizing the unified first law of thermodynamics alongside the Kodama-Hayward temperature, we derive a generalized set of Friedmann equations characterized by a fractional parameter $\alpha \in (1,2]$. The thermodynamic analysis reveals that the specific heats $C_V$ and $C_p$ share the same sign and depend solely on the deceleration parameter, demonstrating that the fractional model is thermodynamically stable during the late-time accelerated expansion and does not exhibit phase transitions. To constrain the background dynamics, we confront the truncated fractional model with a joint sample of late-time observational data, including Cosmic Chronometers, Pantheon+SH0ES supernovae, and the latest DESI DR2 Baryon Acoustic Oscillations. Exploring the physically motivated range $1 < \alpha \leq 2$, we find that the fit quality degrades monotonically as $\alpha$ decreases from the General Relativity limit, with the data favoring $\alpha$ close to $2$ while yielding $H_0 = 69.50 \pm 0.42$ km/s/Mpc and $\Omega_{m0} = 0.292 \pm 0.008$ at $\alpha = 2$. Decreasing $\alpha$ coherently shifts $H_0$ upward and $\Omega_{m0}$ downward, revealing that the fractional parameter modulates the background expansion in a physically nontrivial and observationally distinguishable way.