Researchers have explored two late-time cosmological models derived from Kruglov's nonadditive entropy applied to the apparent horizon. These models aim to explain the dynamics of cosmic expansion and dark energy, an enigmatic component driving the universe's acceleration. The main difference between them lies in how the generalized horizon entropy is incorporated into the Friedmann equations, which describe the cosmos's evolution.

Model I, a previous derivation by Kruglov, uses the Cai-Kim thermodynamic construction and results in an implicit modified Friedmann equation. Model II, proposed in this work, is obtained from an entropy-integral prescription and contains an effective vacuum contribution. Both models were formulated in terms of the dimensionless Hubble parameter, including radiation, and exact closure for the present day was imposed. The physical branch's evolution was performed using analytical derivatives of the implicit Friedmann equations.

The models were constrained with observational data, including baryon acoustic oscillations (BAO) from DESI DR2 and a compressed CMB likelihood, with and without DES-Y5 Type-Ia supernovae data. Model I predicts a significantly lower Hubble constant (H₀) of approximately 62.4 km s⁻¹ Mpc⁻¹, and a dark energy equation of state parameter (wDE,0) of -0.814, making it strongly disfavored compared to the flat ΛCDM model. In contrast, Model II stays close to standard ΛCDM cosmological parameters, with a wDE,0 of -1.003, although its two entropy-sector parameters show strong degeneracy. Its best-fit likelihood is comparable to ΛCDM, but information criteria do not support the additional parameters. This highlights that the observational plausibility of entropy-based cosmology critically depends on how horizon entropy is implemented in Friedmann dynamics.