  We study the von Neumann and R\'enyi entanglement entropy (EE) of
scale-invariant theories defined on tori in 2+1 and 3+1 spacetime dimensions.
We focus on spatial bi-partitions of the torus into two cylinders, and allow
for twisted boundary conditions along the non-contractible cycles. Various
analytical and numerical results are obtained for the universal EE of the
relativistic boson and Dirac fermion conformal field theories (CFTs), and for
the fermionic quadratic band touching and the boson with $z=2$ Lifshitz
scaling. The shape dependence of the EE clearly distinguishes these theories,
although intriguing similarities are found in certain limits. We also study the
evolution of the EE when a mass is introduced to detune the system from its
scale-invariant point, by employing a renormalized EE that goes beyond a naive
subtraction of the area law. In certain cases we find non-monotonic behavior of
the torus EE under RG flow, which distinguishes it from the EE of a disk.
