  We consider conformal field theories in 1+1 dimensions with W-algebra
symmetries, deformed by a chemical potential \mu for the spin-three current. We
show that the order \mu^2 correction to the Re'nyi and entanglement entropies
of a single interval in the deformed theory, on the infinite spatial line and
at finite temperature, is universal. The correction is completely determined by
the operator product expansion of two spin-three currents, and by the
expectation values of the stress tensor, its descendants and its composites,
evaluated on the n-sheeted Riemann surface branched along the interval. This
explains the recently found agreement of the order \mu^2 correction across
distinct free field CFTs and higher spin black hole solutions holographically
dual to CFTs with W-symmetry.
