  Expanding the work of arXiv:1504.08040, we show that black holes obey a
second law for linear perturbations to bifurcate Killing horizons, in any
covariant higher curvature gravity coupled to scalar and vector fields. The
vector fields do not need to be gauged, and (like the scalars) can have
arbitrary non-minimal couplings to the metric. The increasing entropy has a
natural expression in covariant phase space language, which makes it manifestly
invariant under JKM ambiguities. An explicit entropy formula is given for
f(Riemann) gravity coupled to vectors, where at most one derivative acts on
each vector. Besides the previously known curvature terms, there are three
extra terms involving differentiating the Lagrangian by the symmetric vector
derivative (which therefore vanish for gauge fields).
