Second law of horizon thermodynamics during cosmic evolution

We examine the second law of thermodynamics in the context of horizon cosmology, in particular, whether the change of total entropy (i.e. the sum of the entropy for the apparent horizon and the entropy for the matter fields) proves to be positive with the cosmic expansion of the universe. The matter fields inside the horizon obey the thermodynamics of an open system as the matter fields has a flux through the apparent horizon, which is either outward or inward depending on the background cosmological dynamics. Regarding the entropy of the apparent horizon, we consider different forms of the horizon entropy like the Tsallis entropy, the R\'{e}nyi entropy, the Kaniadakis entropy, or even the 4-parameter generalized entropy; and determine the appropriate conditions on the respective entropic parameters coming from the second law of horizon thermodynamics. The constraints on the entropic parameters are found in such a way that it validates the second law of thermodynamics during a wide range of cosmic era of the universe, particularly from inflation to radiation dominated epoch followed by a reheating stage. Importantly, the present work provides a model independent way to constrain the entropic parameters directly from the second law of thermodynamics for the apparent horizon.