In general the plugin works fine on a multisite. Due to the fact that there is no network-wide functionality, please make sure that enable it on each sub site you want to use it on.
All your sites will have its own settings. That's why you have to enter your license key on the plugin settings page of every single blog inside your multisite.
Which license do I need?
In order to use AAWP on a WordPress multisite, you must own at least a “Plus” license (see pricing).
Each site on a WordPress multisite installation counts as a separate site.
In case you're license is does not match your multisite intention, you can easily upgrade your license.
Plugin updates in a multisite
For technical reasons, new plugin updates only appear if the plugin is active in the first blog (ID 1) of your multisite.