Perpetual indexing shouldn't be a thing. With me indexing only happens on changes in pom.xml or project.clj files. Could there be anything that triggers file touches in one of those ?
Or is it perhaps not indexing but do you have a Mercurial project and is the option 'Check incoming and outgoing changes' on in the VCS > Mercurial settings ? It should say it's doing that in the status bar when active though, it's not the same as indexing.
That's a completely optional feature, which can interrupt editing in Cursive and kicks in quite often. My guess is it happens with problems connecting to the upstream repo, timing out and retrying ad infinitum.
That happened to me when it tried to connect to a local domain dns name while I was outside the domain. I disabled it for that reason, since I'm not bothered about incoming changes at all.