  For the typical quantum many-body systems that obey the eigenstate
thermalization hypothesis (ETH), we argue that the entanglement entropy of
(almost) all energy eigenstates is described by a single crossover function.
The ETH implies that the crossover functions can be deduced from subsystem
entropies of thermal ensembles and have universal properties. These functions
capture the full crossover from the groundstate entanglement regime at low
energies and small subsystem size (area or log-area law) to the extensive
volume-law regime at high energies or large subsystem size. For critical
one-dimensional systems, a universal scaling function follows from conformal
field theory (CFT) and can be adapted for nonlinear dispersions. We use it to
also deduce the crossover scaling function for Fermi liquids in $d>1$
dimensions. The analytical results are complemented by numerics for large
non-interacting systems of fermions in $d=1$, $2$, and $3$ dimensions and have
also been confirmed for bosonic systems and non-integrable spin chains.
Furthermore, one can deduce the distribution function for eigenstate
entanglement.
