  In this letter, 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 $S_n = \gamma
\frac{n}{1+n}\frac{2 \pi\,k_B\,c^3}{G\,\hbar} L^{n+1}$ 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 analyse 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 $\Lambda$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
$\Lambda$CDM from which, in Bayesian terms, our models are indistinguishable.
