Whenever I play the cello, my hand tends to get sore the longer I hold my bow. I try to bend my thumb, but whenever I do, the bow slips.
There may be multiple problems with your grip. You didn't say whether you're taking lessons or not, so I'll just emphasize that you need a teacher or an experienced cellist to look at your grip and make sure it's correct. You want most of the pressure (and not much at that!) to be applied with your first and pinky fingers while the other two provide stability. The thumb should just help balance against the fingers. I know teachers who use the "eensy weensy spider" exercise: hold the bow vertically and practice manipulating it with just the thumb, forefinger, and ring finger to "climb" up and down the length of the bow.
Beyond fingers, the wrist and elbow positions are critical as well. You want the weight of your arm to help set the bow, rather than trying to do everything with the fingers. If you can find a copy of Chris Bunting's books ,grab that-- there are a lot of diagrams and discussions of hand and body positions.
And finally, when you feel tired or cramped, stop and rest. Shake your hand loose. Keep things as relaxed as possible to avoid the risk of long-term injury.
Carl Witthoft has a good answer, here are just a few complements:
Try to think about your bow as a pen for drawing or writing. If you hold it right, with the best angle and position, moving it and producing sound should feel almost effortless, guiding it, not pushing or pulling forcefully. There are people holding their pen very tight but they end up writing quite slowly and not very legibly.
Are you fingertips mostly on the stick, or are they in contact with the frog (the (usually dark wood) block behind the stick) ? Check that at least your middle and ring finger reach down to the frog so that you can use this support to guide the bow.
Do you apply a reasonable amount of rosin on the hair of your bow ? Insufficient quantity of rosin will make you want to apply more pressure to produce sound, and in response one tends to hold one's bow quite too strongly. (Also take time to clean your strings after each working session).
What is the slope between your instrument and the ground ? You will find producing a large sound is often easier when the cello is more horizontal so that gravity helps you.
Try with different tension of your bow's hair and see if you can manage to release some pressure on the bow while moving it. Practice releasing some pressure on the bow once you have started producing sound, and do that both on open strings and on stopped strings (the sensation and result is quite different).