You can find out more details about how to qualify for access to the lounge. But they do not appear to allow for one time fees for entry when you do not qualify. Also, the first class/business tickets usually must be international, with a few exceptions.
It is free if you are in first class or business class (we were in business class). You can learn more about accessing it at their website: https://www.aa.com/i18n/travel-info/clubs/flagship-lounge.jsp
It is free if you are in first class
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