  We propose a definition for topological order at nonzero temperature in
analogy to the usual zero temperature definition that a state is topologically
ordered, or "nontrivial", if it cannot be transformed into a product state (or
a state close to a product state) using a local (or approximately local)
quantum circuit. We prove that any two dimensional Hamiltonian which is a sum
of commuting local terms is not topologically ordered at $T>0$. We show that
such trivial states cannot be used to store quantum information using certain
stringlike operators. This definition is not too restrictive, however, as the
four dimensional toric code does have a nontrivial phase at nonzero
temperature.
