  We investigate the stability of spherically symmetric, purely magnetic,
soliton and black hole solutions of four-dimensional ${\mathfrak {su}}(N)$
Einstein-Yang-Mills theory with a negative cosmological constant $\Lambda $.
These solutions are described by $N-1$ magnetic gauge field functions $\omega
_{j}$. We consider linear, spherically symmetric, perturbations of these
solutions. The perturbations decouple into two sectors, known as the
sphaleronic and gravitational sectors. For any $N$, there are no instabilities
in the sphaleronic sector if all the magnetic gauge field functions $\omega
_{j}$ have no zeros, and satisfy a set of $N-1$ inequalities. In the
gravitational sector, we are able to prove that there are solutions which have
no instabilities in a neighbourhood of stable embedded ${\mathfrak {su}}(2)$
solutions, provided the magnitude of the cosmological constant $\left| \Lambda
\right| $ is sufficiently large.
