This is a little Sphinx extension that does exactly that. It also checks for
included files and other dependencies and uses their "last updated" time if it's
more recent. For each file, the "author date" of the Git commit where it was
last changed is taken to be its "last updated" time. Uncommitted changes are
ignored.

If a page doesn't have a source file, its last_updated time is set to None.

The default value for html_last_updated_fmt is changed from None to the empty
string.