Extended Entropic Dark Energy with Four Free Parameters: Theory, Dynamics, and Constraints

We investigate a four-parameter entropic dark energy model in a spatially curved FLRW universe, based on a generalized entropy-area relation at the apparent horizon. While the proposed entropy function captures a broad class of gravitational entropy corrections, including Bekenstein-Hawking, Tsallis, and power-law forms, it does not encompass information-theoretic entropies such as Sharma-Mittal or Renyi. Within this framework, we derive exact analytical expressions for key cosmological observables, including the Hubble parameter $H(z)$, the dark energy density parameter ${\mathrm{\Omega }}_D(z)$, and the equation of state $w_D(z)$. A comprehensive parameter-space analysis reveals viable regions, particularly for $\beta >1$ and small positive curvature, that accommodate elevated $H_0$ values consistent with recent SH0ES measurements. Our results offer a simple and analytically tractable alternative to conventional dynamical dark energy models, with potential relevance to the ongoing Hubble tension.