In practice, cool URLs can become inaccessible even if they don't change
The idea that "cool URLs don't change" has been an article of faith for a very long time. Of course this is false in practice; decades of experience have shown us that cool URLs do change for all sorts of reasons (for example, forced changes in your domain name). Recently, we've seen at least two examples where cool URLs may not change as such, but they do become broadly inaccessible.
The first case is Reddit, where people's protests against recent corporate moves have caused a significant number of communities to become 'private' (for the moment), which makes them inaccessible to both the general Internet and most logged in Reddit users. This is a relatively extreme case because it's pretty close to a soft deletion of the content. A second and more illustrative case is that within the past day or so, Twitter has stopped letting you see anything (tweets, profiles, etc) unless you're a logged in user of the site. It's possible that this is temporary and the situation will change back to the relatively open access that it used to be, but it's also possible that going forward, Twitter's 'content' will only be accessible to registered, logged in (and allowed) users of Twitter.
Of course, partially or completely restricting access to 'content' so that you have to be a registered, logged in user is something that has a long history on the web. Some websites have always been that way, with the goal of increasing the number of registered users they have. Others have started out open to attract people and then closed down access later to drive registration and often 'monetization'. Twitter (and Reddit) are mostly notable for how long they've been open access, and thus how many casual references to things they host are out there on the Internet.
(I have plenty of entries with links to tweets, all of which are now relatively inaccessible (if they still exist; many probably don't).)
I'm using 'cool URL' in a somewhat loose sense here, because Reddit and Twitter never promised that these URLs would be eternal. But almost no one does, almost everyone tries to preserve URLs, and these URLs have been stable for what is often a very long time. In practice, 'open' websites like the old Reddit and Twitter generally have an incentive to keep their URLs cool and unchanged, because it encourages people to link to them, which drives traffic. If you couldn't link to or embed tweets, Twitter might have been much less pervasive than it became for a while.
PS: There are various reasons to not register on a website merely to see things on the website, including that registering usually requires you to agree to their terms of service, which often contain things ranging from bad to terrible.