Windows update error 80072efd is an error occurring while checking updates. The most probable cause to this error is a number of update requests that is too high. You can try to solve this problem by closing Windows Update and waiting for a fifteen minutes or so. Then open Windows update and try running it again.
If the problem persists, it may be an indicator that there is a program on your computer that is still running, so Windows update service is unable to perform its task. The programs causing this error are anti-spywares, firewall, anti-virus softwares, proxy servers and web accelerators.
You can try adding the Windows update website to the list of firewall exceptions. If the problem is still there, go to the support website and try to find a solution. Here is another thing you can do: see if you are connected to the Internet first, since you need the Internet for downloading the updates. Turn the firewall off. You have to turn all firewalls off, meaning Windows firewall, too. Turn off all anti-virus and anti-spyware programs. Also, turn off web accelerators, since they can also cause this kind of problems.
You will have to add the Microsoft update website and Windows update website to the list of trusted websites. If you continue to have problems, try to reboot the server. Log in as an administrator and run Microsoft updates. Go to HOSTS file and open %windir%\system32\drivers\etc\HOSTS text in notepad to check if there are entries for Windows updates. Then reset the proxy server. Type netsh winhttp reset proxy and check the settings of group policy. Some of these settings can lead to errors.
Those are some steps you can try in order to solve this issue. If Windows update error 80072efd message continues to show up on your screen and you just can not fix it, call a professional and ask for help.