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Recently I've been reading T.H.Colding's paper of Ricci curvature and volume convergence. A proof of the continuity of volume functions was given under the lower Ric bounded condition. Having searched online with failure, I wonder if there were some results of other quantity's regularity under the Gromov-Hausdorff topology.

P.S. I want to find some other properties that could change like the volume of manifolds with a lower Ric bound under the GH-topology.

Thanks

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Some metric quantities like diameter, radius or maximal packing converge by trivial reasons. But it seems that you are interested in integral quantities. Here is one result from our paper with Nina Lebedeva:

Suppose $M_1,M_2,\dots$ is a converging and noncollapsing sequence of closed $m$-dimenisonal Riemannian manifolds with lower sectional curvature bound. Then the sequence $$s_n=\int\limits_{M_n}\mathrm{Sc}$$ converges. Here $\mathrm{Sc}$ stands for scalar curvature.

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