How to I list applications sorted by modified time in Finder on OS X 10.8? I discovered that on 10.8 the Applications folder does not allow you to change the sort order, it's always listed by name.

How can I change the sort order?
The view your screenshot displays can be produced if you go to the View menu and select Arrange by ▸ Name (or press ^⌘1). When "Arrange by Name" is selected, indeed changing the sort order is not possible. If, however, you select View then Arrange by ▸ None or press ^⌘0, your problem is solved.