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We propose a new five-parameter entropy function that proves to besingular-free during the entire cosmic evolution of the universe, and at thesame time, also generalizes the Tsallis, Barrow, R\'{e}nyi, Sharma-Mittal,Kaniadakis and Loop Quantum Gravity entropies for suitable limits of theparameters. In particular, all the above mentioned known entropies becomesingular (or diverge) when the Hubble parameter vanishes in course of theuniverse's evolution (for instance, in bounce cosmology at the instant ofbounce), while the newly proposed entropy function seems to be singular-freeeven at $H = 0$ (where $H$ represents the Hubble parameter of the universe).Such non-singular behaviour of the entropy function becomes useful indescribing bouncing scenario, in which case, the universe undergoes through $H= 0$ at the instant of bounce. It turns out that the entropic cosmologycorresponding to the singular-free generalized entropy naturally allowssymmetric bounce scenarios, such as -- exponential bounce and quasi-matterbounce scenario respectively. In the case of exponential bounce, theperturbation modes are in the super-Hubble domain at the distant past, whilefor the quasi-matter bounce, the perturbation modes generate in the deepsub-Hubble regime far before the bounce and hence resolves the ``horizonissue''. Based on this fact, we perform a detailed perturbation analysis forthe quasi-matter bounce in the present context of singular-free entropiccosmology. Furthermore the entropic cosmology in the present context is shownto be equivalent with the generalized holographic cosmology where theholographic cut-offs are determined in terms of either future horizon and itsderivative or the particle horizon and its derivative.

A non-singular generalized entropy and its implications on bounce cosmology