  For general metric theories of gravity, we compare the approach that
describes-derives the field equations of gravity as a thermodynamic identity
with the one which looks at them from entropy bounds. The comparison is made
through the consideration of the matter entropy flux across (Rindler) horizons,
studied by making use of the notion of a limiting thermodynamic scale $l^*$ of
matter, previously introduced in the context of entropy bounds.
  In doing this: i) a bound for the entropy of any lump of matter with a given
energy-momentum tensor $T_{ab}$ is considered, in terms of a quantity which is
independent of the theory of gravity we use; this quantity is the variation of
the Clausius entropy of a suitable horizon when the element of matter crosses
it; ii) by making use of the equations of motion of the theory, the same
quantity is then expressed as the variation of Wald's entropy of that horizon
(and this leads to a generalized form of the generalized covariant entropy
bound, applicable to general diffeomorphism-invariant theories of gravity);
iii) a notion of $l^*$ for horizons, and an expression for it, is given.
