First, I feel your pain. I'm in Houston at my girlfriend's place; we live in a single family house, and the A/C is running all day even with the thermostat at 84 degrees. I work from home, so I can't let it go up higher than that. Our outside thermometer recorded 106 degrees today.
The things that we've added to the house to help keep cool, besides new insulation and having the A/C unit checked, include a bunch of things. First, a little theory. Your apartment heats up because of the sun heating the building primarily, and secondarily because of infiltration from the outside of warm air.
I'm going to just assume that your landlord isn't willing to replace your windows and doors with low-E dual-paned units. I've done this to my house in Bryan, TX and it literally cut my utility bill in half.
The first thing we did with her place, where the landlord is also not willing to rip out a ton of windows and replace them, was to get blinds with reflective shades in all of the windows, and external solar shades or screens where possible. Remember: Black stuff is OK on the outside of the structure, because black absorbs heat (and you don't care about the heat if it's on the outside) -- and white things on the inside of the structure, because you want the heat to not be absorbed and to just be reflected back out.
The second thing we did was to get a better thermostat for the house. The old one was a manual one with mercury in it; the new one has a digital timer and a bunch of other features that include breaks for things like the compressor coils to defrost.
The third thing we did was start replacing the A/C filter frequently... at least once a month.
It might help to suggest more if we knew what kind of A/C unit you have (is it one of those high-rise below-window units, or a 'real' one with a compressor and air exchanger in a closet or attic?), and what kind of structure (concrete or wood frame) you're in.
I ended up pulling the capacitor and testing it with my ohm meter. It was dead. I could have easily gone to the local HVAC supply, purchased a new capacitor for $30 and installed it myself. But, because I also wanted to have the pressure tested, I called in the guy who installed my furnace. He affirmed that I "did good", it was the capacitor. And because I had done most of the diagnostic work, he cut me a good deal.
In the end, the most important bit, on a 95 degree day when the humidity is 70%, a happy wife is a blissful life. ;)
Best Answer
From inside your apartment, no, the best you can do is use a fan, open the windows, block the sun with blinds, or walk around naked. And if you do that last part in the building, the management may decide to turn on the AC sooner.