Depends. Some do and some don't. If you use a DC pump when you turn off the computer you will turn off the pump. If you have an AC pump then things get more fun.
My pump is plugged directly but has a switch I mounted just below the power switch on my comp. I turn it on at the same time as my comp and leave it running a few min usually when i turn the comp off. I have left it running overnight/all day before. Its not a problem.
P.S.Quite a few pumps are rated for 50,000+ hours of continuous use. Thats over 5 1/2 years.
if your pump is rated to run all the time, run it all the time if you're lazy. if it's not supposed to be run all the time or doesn't have a good life expectancy i would be more reluctant. alternatively you could get a relay switch, it's what i use for mine.