It could be the literal sense after all.
I guess the usual doors might let out too much of the refrigerated air from the building, whereas the revolving doors might help keep the refrigerated air inside.
I cite from The Rockefeller University website:
Seal drafty windows and doors with insulation kits, caulk or weather-stripping to keep the warm air inside. It also helps to remove window air conditioners, or to cover up the interior and exterior faces of permanent ones. When on campus, make sure to use the revolving doors to prevent heat from escaping — they save up to 30% in energy costs spent reheating lobbies.
In a hot climate, the same might apply to preventing cool air from escaping.
Here's a weighty treatise on the energy-conserving effect of revolving door use (Massachusetts Institute of Technology).
P.S. Why did the authors of the sign choose the wording
Please use the revolving doors to keep you cool
Please use the revolving doors to keep the building cool
is anybody's guess: I guessed it's an attempt at wordplay, but ColleenV and Jay say (see comments below) it is likely the means to personalize the suggestion, to make people understand that it's in their own interest to use the revolving door.