  We analyze the entanglement entropy, in real space, for the higher
dimensional integer quantum Hall effect on ${\mathbb {CP}}^k$ (any even
dimension) for abelian and nonabelian magnetic background fields. In the case
of $\nu=1$ we perform a semiclassical calculation which gives the entropy as
proportional to the phase-space area. This exhibits a certain universality in
the sense that the proportionality constant is the same for any dimension and
for any background, abelian or nonabelian. We also point out some distinct
features in the profiles of the eigenfunctions of the two-point correlator that
underline the difference in the value of entropies between $\nu=1$ and higher
Landau levels.
