Assuming the thermostat is good, since it was supposedly replaced, and the coolant level is adequate, I would look at the water pump. You car might be one with the plastic impeller on the water pump. Over time, this plastic impeller will break away from the shaft. Eventually, it'll lock up in the bore, and not pump at all.
What can happen is this. The impellar will get loose on the shaft, but not broken. At slow engine speed, the impeller will spin fine, and pump water. As you increase the rpm, the resistance of the coolant will cause the impellar to spin on the shaft, and not pump. Once it breaks loose and starts spinning, it won't pump again until the rpms come back down.
You can check the output this way: with the engine cool or warm, not hot, pull offf the radiator cap, carefull of overflow.
Run the engine at idle, and look for coolant flow from the bleeder port where the cap screws in. It may or may not be pumping. Rev the engine up a little, and see if you get flow. If you do, rev it up more and see if the flow stops.
If the flow stops, or isn't there at all, the pump is probably not pumping.
The system needs to be full and bled for this test to work. Sometimes at idle, there won't be much flow indicated, even though there is nothing wrong. If you rev it up, there should be a good stream.
Also check to make sure the clutch fan is ok, and the auxillary fan.