I found an answer directly from a style guide. According to the APA Blog (6th edition),
[W]e usually don’t ask, “When did you consult that source?” One exception to this rule would be for material that is subject to frequent change, such as Wikipedia entries. Because this information is designed to be constantly updated, it’s important to let readers know when you retrieved it.
You can check the revision history of Wikipedia (though usually a poor academic source) but it isn't the only site that has a public record like that. Stack Exchange sites do too, as do many software documentation sites (through GitHub). Still other sites have private revision histories that an admin has access to. (For example, WordPress sites.)
Another answer brought up archival sites like archive.org. If the page wasn't updated, but the URL no longer works, then most likely you will only need to check one entry to find the information: the first. (You may also be able to search the internet instead to see if the page moved by using the other details, namely title and author.) But with a frequently updated source, it's possible that the cited information only appeared in a single revision of the page. Without knowing which, you'd have to check every snapshot (excluding those after the citation was published). And with an external archival site, the page could have changed between snapshots, so it's a good idea to save the page yourself if you're citing it. However, the comments on the APA article indicate that they believe archival sources to be secondary sources, so the URL of the original is what should go in the citation list.