From Patch 1.20 Notes, MIND and PIETY are the stats that affect damage dealt by thaumaturge arms and grant Auto-attack Damage Bonus.
Attack Power is related to STRENGH and damage dealt by pugilist, gladiator, marauder, and lancer arms.
My thought is Attack Power doesn't have a noticeable effect for THM. It doesn't have any effect, indeed.

PS: Is there any trick on this question? u.u'