  The holographic cut-off in generalized dark energy (HDE) formalism depends on
its cut-off. Following this, a four-parameter generalized entropy has recently
been developed. It reduces to various known entropies for appropriate parameter
limits in the study of Odintsov, S. D., S. DOnofrio, and T. Paul. (2023)
Physics of the Dark Universe, 42 pp: 101277. In the current work, we
investigate the evolution of the universe in its early phase and late phase
within the framework of entropic cosmology, where the entropic energy density
functions are reconstructed within the framework of the equivalence of
holographic dark energy and four-parameter generalized entropy (Sg). Along with
the reconstruction as mentioned earlier scheme, in this study, we demonstrate
that an extensive variety of dark energy (DE) models can be considered distinct
and particular candidates for the most generalized four-parameter entropic HDE
family, each having their cut-off. We examined several entropic dark energy
models in this regard, including the generalized holographic dark energy with
Nojiri-Odintsov(NO) cut-off, the Barrow entropic HDE (BHDE) with particle
horizon as IR cut-off, the Tsallis entropic HDE (THDE) with future event
horizon as IR cut-off, all of three cases are particular cases of the most
generalized four parameter entropic holographic dark energy. Inspired by S.
Nojiri, and S. D. Odintsov (2006) (General Relativity and Gravitation, 38 p:
1285-1304 ) and (S. Nojiri and S. D. Odintsov, 2017, European Physical Journal
C, 77, pp.1-8 ); our current work reports a study on cosmological parameters
and thermodynamics with entropy-corrections (logarithmic and power-law) to
cosmological horizon entropy as well as black hole entropy with a highly
generalized viscous coupled holographic dark fluid along its particular cases.
