  Stable light rings, which are associated with spacetime instabilities, are
known to exist in four-dimensional stationary axisymmetric spacetimes that
solve the Einstein-Maxwell equations (so-called electrovacuum solutions, with
Faraday tensor $F_{\mu \nu} \neq 0$); however, they are not permitted in pure
vacuum ($F_{\mu \nu} = 0$). In this work, we extend this result to spacetimes
with a non-zero cosmological constant $\Lambda$. In particular, we demonstrate
that stable light rings are permitted in $\Lambda$-electrovacuum ($F_{\mu \nu}
\neq 0$, $\Lambda \neq 0$), but ruled out in $\Lambda$-vacuum ($F_{\mu \nu} =
0$, $\Lambda \neq 0$).
