  We consider arbitrary stationary and axisymmetric black holes in general
relativity in $(d +1)$ dimensions (with $d \geq 3$) that satisfy the vacuum
Einstein equation and have a non-degenerate horizon. We prove that the
canonical energy of axisymmetric electromagnetic perturbations is positive
definite. This establishes that all vacuum black holes are stable to
axisymmetric electromagnetic perturbations. Our results also hold for
asymptotically deSitter black holes that satisfy the vacuum Einstein equation
with a positive cosmological constant. Our results also apply to extremal black
holes provided that the initial perturbation vanishes in a neighborhood of the
horizon.
