Lighting – What type of exterior floodlight will turn on slowly (or not at all) when it’s cold out

The product description on Amazon is pulled directly from the manufacturer's data sheet (the link is an excerpt from a catalog, scroll down two pages for the light fixtures you bought), so I think you can trust it.

Another assurance that you'll be OK: I see from that catalog that the base of the light fixture is 4 1/8" in diameter, which is the same as the outside dimension of the diagonal of a single gang box (at least the one I picked out of my big box of electrical goodies :). When you open the packaging, you should see a rectangular pattern in the molding on the back where the electrical box will be mounted.

1. Confirm your wall switch is good. Bad switches can be intermittent.
2. Turn off the switch and take the tubes back out and put them back in making sure they're fully snapped in. It's worth a try. Don't go buy more, but if you have other new tubes try them.
3. If the lights work for a period of time and go out on their own, that usually indicates an over-heating ballast - replace the ballast.
4. Otherwise, the best test for a ballast is just what you did, try new tubes. If no luck, replace the ballast.
5. While replacing the ballast, make sure all of the wires in the sockets are well connected. Give them each a little tug. If you can't get a wire to stay in a socket replace the socket, maybe replace all 4.
6. Fluorescent lights need to be grounded to work properly. Make sure a ground wire is attached to the fixture. I've seen fluorescent lights, especially in old homes, where the people say they're always replacing tubes, and it's because the fixture isn't grounded.
7. If you're going to replace the ballast or the entire fixture consider going with an electronic ballast maybe even T8 size tubes. Amazon sells the smaller clips you'd need for a retrofit to T8 U-Bent tubes - ASIN B00A3WGR0W.