This oh-so descriptive error basically means the computer can’t find (DNS) or get to (LAN/internet connection) the Windows Update Server, be it Microsoft or in an office environment possibly a local WSUS server.
Windows could not search for new updates.
An error occured while checking for new updates for your computer.
Code 80072EE2 Windows Update encountered an unknown error.
I had this error after updating my router and initially couldn’t figure out why it was related… until I remembered I’d set up a local Windows Update (WSUS) server and set the related Group Policy to update from a wsus.[my domain] alias. This is set in:
Computer Configuration; Policies; Administrative Templates; Windows Components; Windows Updates; Specify intranet Microsoft update service location.
I hadn’t reconfigured this alias so all my computers were failing to resolve the address (DNS) and therefore couldn’t update.
The fix for me was to re-instate the alias but hopefully this might help others to find out what their particular problem is.