  In this article we consider the continuity of the eigenvalues of the
connection Laplacian of $G$-connections on vector bundles over Riemannian
manifolds. To show it, we introduce the notion of the asymptotically
$G$-equivariant measured Gromov-Hausdorff topology on the space of metric
measure spaces with isometric $G$-actions, and apply it to the total spaces of
principal $G$-bundles equipped with $G$-connections over Riemannian manifolds.
