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Because that is the cooled water hose. Water only flows through the lower hose once the thermostat has opened. If the thermostat is closed there is no or limited flow through the radiator. If the outside temp is cold it may take a while to be able to heat up the water in the radiator to be able to notice it in the lower hose.
The hot water flows from the upper hose into the radiator and then it is cooled, the lower hose normally will not show heat until the car has been run for a while or under a load.
The best way to see water circulation is to use a laser thermometer. This you can point at the radiator and follow the temp of the coolant. I dont think you have an issue with the thermostat or the water pump. Running the car that long it would overheat if there was an issue with the water pump and or thermostat. The aluminum radiators in these cars are very efficient, so without knowing the ambient temp its hard to determine why the lower hose is cold, other then the radiator is efficient and cooling the coolant going through it
I have seen many of these at the dealer were the coolant flow is very restricted on the heater core.
You can also use a scanner to see what the coolant temp is in the block, but that normally requires a factory type scanner or the Ross-Tech program.
That temp is higher than what it should be. The turbo does not help with the coolant flow and there is no additional water pump that works with the car running. The only item that works is the water pump driven off the timing belt. Did you use a factory thermostat or after market. Also did you vacuum fill the car or just fill the coolant tank.
This car if equipped with an after run pump, only circulates a small amount of coolant after the car is shut down. The after run pump do not work when the car is running. Not all are equipped with this feature.