  We continue to explore the connections between large deviations for objects
coming from random matrix theory and sum rules. This connection was established
in [17] for spectral measures of classical ensembles (Gauss-Hermite, Laguerre,
Jacobi) and it was extended to spectral matrix measures of the Hermite and
Laguerre ensemble in [20]. In this paper, we consider the remaining case of
spectral matrix measures of the Jacobi ensemble. Our main results are a large
deviation principle for such measures and a sum rule for matrix measures with
reference measure the Kesten-McKay law. As an important intermediate step, we
derive the distribution of canonical moments of the matrix Jacobi ensemble.
