  Working in the setting of i.i.d. last-passage percolation on $\mathbb{R}^D$
with no assumptions on the underlying edge\hyp{}weight distribution, we arrive
at the notion of grid entropy - a Subadditive Ergodic Theorem limit of the
entropies of paths with empirical measures weakly converging to a given target,
or equivalently a deterministic critical exponent of canonical order statistics
associated with the Levy-Prokhorov metric. This provides a fresh approach to an
entropy first developed by Rassoul-Agha and Sepp\"al\"ainen as a large
deviation rate function of empirical measures along paths. In their 2014 paper
arXiv:1202.2584, variational formulas are developed for the
point-to-point/point-to-level Gibbs Free Energies as the convex conjugates of
this entropy. We rework these formulas in our new framework and explicitly link
our descriptions of grid entropy to theirs. We also improve on a known bound
for this entropy by introducing a relative entropy term in the inequality.
Furthermore, we show that the set of measures with finite grid entropy
coincides with the deterministic set of limit points of empirical measures
studied in a recent paper arXiv:2006.12580 by Bates. In addition, we partially
answer a directed polymer version of a question of Hoffman which was previously
tackled in the zero temperature case by Bates. Our results cover both the
point-to-point and point-to-level scenarios.
