No your bottom or lower radiator hose should not be cold when the engine is running and the coolant is flowing through the radiator and radiator hoses.
If your bottom or lower radiator hose stays cold then it means there's a problem with your engines cooling system and you need to get the problem fixed before some major damage occurs to your engine.
Some things that could cause your lower or bottom radiator hose to be cold and not heat up are bad coolant thermostat, clogged radiator, plugged or bad heater core, low engine coolant or even a possible failing water pump that is not producing enough water or coolant pressure throughout the cooling system.
If the bottom radiator hose doesn't heat up after driving for awhile or after letting the engine idle then you likely have a cooling system problem.
The first thing I would check after the engine cooled off is to see if you have enough coolant in the radiator so that it can flow through the top and bottom radiator hoses properly.
If you're still having problems with the bottom or lower radiator hose staying cold then I would take the car or vehicle to a mechanic to have it diagnosed and fixed.