  We study the mixed-state entanglement structure of chaotic quantum many-body
systems at late times using the recently developed $\textit{equilibrium
approximation}$. A rich entanglement phase diagram emerges when we generalize
this technique to evaluate the logarithmic negativity for various universality
classes of macroscopically thermalized states. Unlike in the infinite
temperature case, when we impose energy constraints at finite temperature, the
phase diagrams for the logarithmic negativity and the mutual information become
distinct. In particular, we identify a regime where the negativity is extensive
but the mutual information is sub-extensive, indicating a large amount of
$\textit{bound entanglement}$. When applied to evaporating black holes, these
results imply that there is quantum entanglement within the Hawking radiation
long before the Page time, although this entanglement may not be distillable
into EPR pairs.
