What type of exterior floodlight will turn on slowly (or not at all) when it's cold out?
Those look like Metal Halide bulbs.  It is normal that they take upwards of 20 minutes to get to full brightness.  They can be affected by cold temperatures, but this could also be a failing ballast or igniter (sometimes part of the ballast).  There could also be some poor electrical connections.

These should not be turned on and off frequently - you have to let them cool for 20min before restarting.