I guess that Google removes already indexed pages that are now returning the HTTP error
Unfortunately I can't find any definitive statement on that matter. Do you know the answer?
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As you know Google will eventually remove pages which return server codes
404 Not Found and
410 Gone, however a 403 is a forbidden response from the server not an error.
Google will most likely continue checking the page and keep it in it's index in the event it becomes accessible. If you would like it to not be indexed set the meta noindex header tag
They get removed from the index. There is no official documentation from Google on this, but you can find plenty of cases where websites have been returning 403 by mistake and got deindexed.