The main Appthemes site including the demo page has been currently restructured and so some features may not have been loaded yet. It may take a couple of days or so but rest assured the Advanced Search feature is still there.
Please help our moderating team work more efficiently by not sending us support questions via PM. You can read more about how AppThemes support works
here. However, you can send a PM to follow up and remind me if I missed your support request/thread.
Thank you and have a nice day.