My car will start to run hot then it will level back out after a while is this the water pump or thermostat

It is difficult to know what is causing your problem without being able to test it, hands on. I have rarely seen a water pump "fail" or have an impeller detach due to corrosion or something like that. The thermostat may be sticking closed too long; but this is something you would want to check using a diagnostic tool to watch the engine temperature. You could have a cooling fan issue also, or some other kind of engine issue.