  The investigation of modifications to the FLRW cosmology resulting from the
consideration of a general entropy for the cosmological apparent horizon is the
subject of this study. Building upon the work of Nojiri and collaborators in
2022, who introduced a class of generalized entropies with four parameters
capable of converging to familiar entropies and addressing specific
cosmological issues, our research explores the impact of correcting the entropy
on the energy-momentum tensor of the cosmic fluid from the outset. Our
calculations demonstrate that, by employing a correction function $f(\rho)$ to
modify the energy-momentum density tensor, the entropic area law
(Bekenstein-Hawking entropy) can still be regarded as a general entropy. The
construction of the function $f(\rho)$ is facilitated through considerations of
the thermodynamics associated with the apparent horizon. Additionally, we
investigate the first and second laws of thermodynamics within this framework
and illustrate how the limitations imposed on the equation of state of the
cosmic fluid can be resolved through the incorporation of this correction
function. Finally, we compute cosmography parameters to analyze the kinematics
of the universe, with particular attention given to the notable influence of
the correction function $f(\rho)$ on these parameters. This paper provides
valuable insights into the application of general entropies to the apparent
horizon of the universe.
