Generalized mass-to-horizon relation: A new global approach to entropic cosmologies and its connection to $\Lambda \mathrm{CDM}$

In this paper we propose a new generalized mass-to-horizon relation to be used in the context of entropic cosmologies and holographic principle scenarios. We show that a general scaling of the mass with the universe horizon as $M=\gamma \frac{c^2}{G}{L}^n$ leads to a new generalized entropy $S_n=\gamma \frac{n}{1+n}\frac{2\pi k_B{c}^3}{G\hslash }{L}^{n+1}$ from which we can recover many of the recently proposed forms of entropies at cosmological and black hole scales and also establish a thermodynamically consistent relation between each of them and Hawking temperature. We analyze the consequences of introducing this new mass-to-horizon relation on cosmological scales by comparing the corresponding modified Friedmann, acceleration, and continuity equations to cosmological data. We find that when $n=3$, the entropic cosmology model is fully and totally equivalent to the standard $\Lambda \mathrm{CDM}$ model, thus providing a new fundamental support for the origin and the nature of the cosmological constant. In general, if $\log \gamma <-3$, and irrespective of the value of $n$, we find a very good agreement with the data comparable with $\Lambda \mathrm{CDM}$ from which, in Bayesian terms, our models are indistinguishable.