  What are the conditions on a field theoretic model leading to a finite
entanglement entropy density? We prove two very general results: 1) Ultraviolet
finiteness of a theory does not guarantee finiteness of the entropy density; 2)
If the spectral dimension of the spatial boundary across which the entropy is
calculated is non-negative at all scales, then the entanglement entropy cannot
be finite. These conclusions, which we verify in several examples, negatively
affect all quantum-gravity models, since their spectral dimension is always
positive. Possible ways out are considered, including abandoning the definition
of the entanglement entropy in terms of the boundary return probability or
admitting an analytic continuation (not a regularization) of the usual
definition. In the second case, one can get a finite entanglement entropy
density in multi-fractional theories and causal dynamical triangulations.
