I think this could also be a good idea for phasing out public APIs -- instead of just taking an API offline, start to fail requests early at low probability, ramping the probability up to 100% over the course of a month or so.
Years ago, during the tail end of the Age of XML, I had a conversation with one of the admins over at W3C.
To make your app work you're supposed to cache all schemas locally but an inordinate amount of people don't. Their apps will be slow because of that, but work anyway. So tons of traffic goes to w3.org. They can't remove those schemas because then nobody can use them the right way (download them 1x and keep the result). But how do you get people's attention?
I pointed out to him that a human downloading the schema isn't going to notice if the request takes .3 seconds or 3 seconds, but someone loading that schema 100x in a loop is sure going to be incentivized to figure out where the extra 5 minutes of processing time came from. He allowed that I might be onto something there but I never heard back about whether he tried it.
You want to kill off an API, assign it resources (hardware, bandwidth) that get progressively smaller, or add artificial waits that get progressively bigger. Depending on how horizontally scalable your app is (is it easier to add a sleep or to pull servers out of the cluster?).
Have a search for libraries in your favorite language, I'm sure something will already exist. I've personally used Polly in .NET.