Most apartments have a kitchen of some sort, and there's a lot to look for here. First, check out the stove. Most stoves and ovens are electric (rather than gas) in apartment complexes. Do all of the switches work, and do all of the burners get hot when you turn them on? Do you see any frayed wiring around the oven? And if you do happen to have a unit with a gas stove, does it ignite properly? Is there a smell of natural gas when it's put to use? If you have a gas stove, definitely invest in a carbon monoxide detector.
Also, take a good look at the refrigerator. Is it cold? Is there any mold in there? Is it clean, and did the last tenant throw out all of the remaining food?
This is also the time to check for any mold or mildew that might be in the kitchen, especially inside cabinets. You should also look for animal droppings that indicate a pest problem.