The Integrity check of the maintenance job is calling the native SQL dbcc checkdb command, which can consume a lot of tempdb space if the database it is running against is very large. Ensure you have an archive (and/or purge) job to help keep the live database growth in check.
If you run the following sql query against the AA database manually, and you see the same failure, the only resolution is to ensure there is sufficient disk space for the tempdb and the log files, as well as trying to reduce the size of the AA database through archiving:
dbcc checkdb with no_infomsgs, all_errormsgs
