Access-Control-Max-Age has, unfortunately a big security caveat which is that it is cached on a per-endpoint basis. Because Access-Control-Allow-Origin only allows one origin specification, if you previously used the Origin header to determine who could access the API, your next API requestor will effectively get your last response.
For example, to allow abc.com AND bcd.com, you could check Origin and if correct return Access-Control-Allow-Origin: *. In this case, setting Access-Control-Max-Age will mean this applies to any site* after a single successful request. If you return the site's name, this will break CORS* for that amount of time if an attacker makes a single request to it.
This is because you have forgotten to return a `Vary: Origin` header in the response. If you don’t do this, caches will presume the response is the same regardless of the Origin header in the request and so you will get the bug you describe.
Of course note that some common CDNs just ignore the `Vary` header. cough Cloudflare cough. Also IIRC not refusing to cache anything with a `Vary` header but caching it anyways and serving it no matter what headers the client sends.
For example, to allow abc.com AND bcd.com, you could check Origin and if correct return Access-Control-Allow-Origin: *. In this case, setting Access-Control-Max-Age will mean this applies to any site* after a single successful request. If you return the site's name, this will break CORS* for that amount of time if an attacker makes a single request to it.