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When accessing a database entry's password, the log shows a recent access time, but the indicator to save the database is not set. This means when closing KeePass (without explicitly saving) those last access times are not saved.
This is the intended behavior.
Most users don't want to save the database each and every time. If you wish KeePass to remember a last access time change (even without any real modifications to the data), you must manually invoke the 'Save' command (which by default you can do even when it's displayed greyed out).