  In this note we explored the holographic relative entropy in the presence of
the 5d Chern-Simons term, which introduces a mixed gauge-gravity anomaly to the
dual CFT. The theory trivially satisfies an entanglement first law. However, to
quadratic order in perturbations of the stress tensor $T$ and current density
$J$, there is a mixed contribution to the relative entropy bi-linear in $T$ and
$J$, signalling a potential violation of the positivity of the relative
entropy. Miraculously, the term vanishes up to linear order in a derivative
expansion. This prompted a closer inspection on a different consistency check,
that involves time-delay of a graviton propagating in a charged background,
scattered via a coupling supplied by the Chern-Simons term. The analysis
suggests that the time-delay can take either sign, potentially violating
causality for any finite value of the CS coupling.
