Consider two subsystems $A$ and $B$.
In a local quantum field theory defined on the composite system $AB$, why is the divergent piece of the entanglement entropy of subsystem $A$ given by a local integral over the entangling surface $\partial A$?
For reference, see equation (19.3) on page 176 of Thomas Hartman's notes on Quantum Gravity and Black Holes.
In general, this isn't true of a highly excited state - it's only true of the ground state. As the notes point out above (19.3), that equation only holds in the context of the ground state. The fact that the entanglement entropy of the ground state of a system scales as the area of the entangling surface $\partial A$ is quite generic. It's been rigorously proven for gapped systems in one dimension (https://www.nature.com/nphys/journal/v9/n11/full/nphys2747.html) and for free scalar fields (https://arxiv.org/abs/hep-th/9303048). This scaling behavior can also be motivated by the AdS/CFT correspondence (https://arxiv.org/abs/hep-th/0603001). See https://arxiv.org/abs/0808.3773 for a review. Be careful though: the scaling is supplemented by a logarithmic correction for gapless 1D systems and systems with a codimention-1 Bose or Fermi surface.
