  We study the universal contributions to the entanglement entropy (EE) of 2+1d
and 3+1d holographic conformal field theories (CFTs) on topologically
non-trivial manifolds, focusing on tori. The holographic bulk corresponds to
AdS-soliton geometries. We characterize the properties of these
regulator-independent EE terms as a function of both the size of the
cylindrical entangling region, and the shape of the torus. In 2+1d, in the
simple limit where the torus becomes a thin 1d ring, the EE reduces to a
shape-independent constant $2\gamma$. This is twice the EE obtained by
bipartitioning an infinite cylinder into equal halves. We study the RG flow of
$\gamma$ by defining a renormalized EE that 1) is applicable to general QFTs,
2) resolves the failure of the area law subtraction, and 3) is inspired by the
F-theorem. We find that the renormalized $\gamma$ decreases monotonically when
the holographic CFT is deformed by a relevant operator for all allowed scaling
dimensions. We also discuss the question of non-uniqueness of such renormalized
EEs both in 2+1d and 3+1d.
