Just got my water pump changed. Now my thermostat is trying to edge up.
Should you change your thermostat when you change your water pump?
You don't need to, but since your coolant system is already drained, it would be a good time to. Standard thermostats are only about $10 and a $2 gasket. Changing your thermostat is typically 2 bolts and a gasket.
I see no reason to do them together unless you have reason to believe the thermostat needs to be replaced, or unless you have an unusual vehicle in which the thermostat is hard to access without taking apart the same things you'd take apart to get to the water pump. Usually the water pump requires taking off the timing belt cover and a lot of other adjacent parts, whereas the thermostat is usually immediately accessible from above or below the engine at the point where the return radiator hose connects.
It depends if you're a pro or a diy type of person. If you're a pro or you're short on time, I totally agree with @rpmerf. As long as you're at it, why not save yourself the bother if it blows later on.
On the other hand, if you're just a do-it-yourself person with no real time constraints if it does blow, there's no real reason to replace it: thermostats don't have a particular life span, and there's no particular reason that you should have to replace it at the same time.