Kruglov-entropy cosmology from apparent-horizon thermodynamics: background dynamics and observational constraints

We investigate two late-time cosmological models generated by Kruglov's nonadditive entropy at the apparent horizon. Model I, originally derived by Kruglov, follows the Cai--Kim thermodynamic construction and leads to an implicit modified Friedmann equation, whereas Model II, proposed and derived in this work, is obtained from an entropy-integral prescription and contains an effective vacuum contribution. We formulate both models in terms of the dimensionless Hubble parameter, include radiation, impose exact present-day closure, and evolve the physical branch using analytic derivatives of the implicit Friedmann equations. We constrain the models with DESI DR2 baryon acoustic oscillations and a compressed CMB likelihood, with and without DES-Y5 Type-Ia supernovae. Model I is driven to a markedly lower Hubble constant, H 0 62.4, km,s-1,Mpc-1 for the combined data, and remains non-phantom with w DE,0-0.814; it is strongly disfavoured relative to flat Λ CDM. Model II stays close to Λ CDM in the standard cosmological parameters and gives w DE,0-1.003, while its two entropy-sector parameters remain strongly degenerate. Its best-fit likelihood is comparable to Λ CDM, but the additional parameters are not supported by information criteria. These results show that the two thermodynamic realizations of the same nonadditive entropy have sharply different phenomenological consequences at the homogeneous-background level. In summary, the observational plausibility of entropy-based cosmology hinges critically on how the generalized horizon entropy is incorporated into the Friedmann dynamics.