  We discuss the field quantisation of a free massive Dirac fermion in the two
causally disconnected static patches of the de Sitter spacetime, by using mode
functions that are normalisable on the cosmological event horizon. Using this,
we compute the entanglement entropy of the vacuum state corresponding to these
two regions, for a given fermionic mode. Further extensions of this result to
more general static spherically symmetric and stationary axisymmetric
spacetimes are discussed. For the stationary axisymmetric Kerr-de Sitter
spacetime in particular, the variations of the entanglement entropy with
respect to various eigenvalues and spacetime parameters are depicted
numerically. We also comment on such variations when instead we consider the
non-extremal black hole event horizon of the same spacetime.
