You're right about the thermal mass of the liquid helping to hold it at a more steady temperature. So, either would work. But the closet is your better bet.
Oh don't give me none more of that Old Janx Spirit
No, don't you give me none more of that Old Janx Spirit
For my head will fly, my tongue will lie, my eyes will fry and I may die
Won't you pour me one more of that sinful Old Janx Spirit