Holographic cosmology with logarithmic equation of state based on a new generalized entropy

The cosmological application of a dark viscous fluid in a spatially flat Friedmann–Robertson–Walker (FRW) universe with a modified logarithmic equation of state (EoS), being an analogue of the equation associated with the isotropic deformation of crystalline solids, is explored. This formalism represents a class of the so-called logotropic fluids, and allows explaining an accelerating late-time universe. Our research is based on a new generalized entropy function proposed by Nojiri–Odintsov–Faraoni [S. Nojiri, S. D. Odintsov and F. Faraoni, From nonextensive statistics and black hole entropy to the holographic dark universe, Phys. Rev. D 105(4) (2022) 044042]. We consider entropic cosmology and describe the evolution of the late time universe in the presence of dark matter coupled with viscous dark energy. In order to obtain a more detailed picture of its evolution, we add in our model a coupling of the log-corrected power-law fluid with dark matter, and study various interacting forms between them. We solve the system of dynamic equations for viscous dark fluid coupled with dark matter and obtain expressions for the log-corrected power-law dark energy density, and the energy density for dark matter. The research results are presented in holographic form in terms of generalized holographic cut-offs introduced by Nojiri and Odintsov [S. Nojiri and S. D. Odintsov, Unifying phantom inflation with late-time acceleration: Scalar phantom-non-phantom transition model and generalized holographic dark energy, Gen. Relativ. Gravit. 38 (2006) 1285; S. Nojiri and S. D. Odintsov, Covariant generalized holographic dark energy and accelerating universe, Eur. Phys. J. C 77 (2017) 528]. In the asymptotic case an expression for the Hubble function, and the energy density for dark matter in the entropic cosmology, are obtained.