  The Bekenstein-Hawking entropy is a cornerstone of horizon thermodynamics but
quantum effects correct it, while inequivalent entropies arise also in
non-extensive thermodynamics. Reviewing our previous work, we advocate for a
new entropy construct that comprises recent and older proposals and satisfies
four minimal key properties. The new proposal is then applied to black holes
and to holographic dark energy and shown to have the potential to cause early
universe inflation or to alleviate the current Hubble tension. We then analyze
black hole temperatures and masses consistent with alternative entropies.
