Base of light bulb turning brown
A compact florescent (CFL) definitely heats up and needs to have room to dissipate that heat.

It is not nearly as hot (25&ndash;33%) as an equivalently bright incandescent, but the radiation pattern is different too.  An incandescent has most of its heat coming out the bulb with only a small proportion from near the narrow base.  A CFL has electronics in its base and those are responsible for a good amount of the bulb's heat.

The different heat pattern would explain why an enclosed base of a CFL would get crispy hot.  If that were combined with stagnant air and warm ambient temperatures then yes, it could be a fire hazard.

However, your pictured light fixture probably does not confine the bulb's bases and was made for&mdash;what? 225 watts of incandescents?  It should be fine for three 40 watt CFLs.

Do this experiment:  fill it full of CFLs at or above the wattage you'd want to run.  Close the cover and turn it on for at least 20 minutes.  Take the cover off and feel inside.  If anything is too hot to touch (120 &deg;F/ 50 &deg;C or greater), then it probably is not safe.  If parts are warm to the touch but not verging on painful, it is most likely fine.

LED bulbs would be completely safe.