The pipes won't be cold but they may not be as hot as what they otherwise normally would be. I.E you feeling them may just be warm, if faulty stat this will cool down with the car in motion I would defo err on the stat, other possibility would be clogged matrix but it's unlikely unless the cooling system has been neglected.

Way to test the stat is on a cold start by feeling the top rad hose, if you can feel coolant running through it then the stat is not doing it's job.

Martin