  We analyze the stability properties of the purely magnetic, static solutions
to the Einstein--Yang--Mills equations with cosmological constant. It is shown
that all three classes of solutions found in a recent study are unstable under
spherical perturbations. Specifically, we argue that the configurations have
$n$ unstable modes in each parity sector, where $n$ is the number of nodes of
the magnetic Yang--Mills amplitude of the background solution. The
``sphaleron--like'' instabilities (odd parity modes) decouple from the
gravitational perturbations. They are obtained from a regular Schr\"odinger
equation after a supersymmetric transformation. The body of the work is devoted
to the fluctuations with even parity. The main difficulty arises because the
Schwarzschild gauge -- which is usually imposed to eliminate the gravitational
perturbations from the Yang--Mills equation -- is not regular for solutions
with compact spatial topology. In order to overcome this problem, we derive a
gauge invariant formalism by virtue of which the unphysical (gauge) modes can
be isolated.
