It's part of the flood detection/prevention. Does this affect approving multiple runs in a row as well, or is this only happening in just this specific case of submitting then editing right afterwards?

It stops you from submitting multiple runs very close to each other, I had some issues when migrating a leaderboard recently to the site. It also prevents you from editing a run very soon after submitting. Just to note that if you click edit soon after submitting but then wait on the edit page for a while it still triggers, so the timer seems to be checked when you load the page, not submit.

Well if I recall correctly, the more you use the website, the smaller the time between two actions is.

At least that is what was discussed when we implemented the flood prevention feature.