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

On spectral convergence of vector bundles and convergence of principal bundles