  In this paper we study the gradient steady K\"ahler-Ricci soliton metrics on
non-compact toric manifolds. We show that the orbit space of the free locus of
such a manifold carries a natural Hessian structure with a nonnegative
Bakry-\'Emery tensor. We generalize Calabi's classical rigidity result and use
this to prove that any complete $\mathbf T^n$-invariant gradient steady
K\"ahler-Ricci soliton with a free torus action must be a flat $(\mathbb
C^*)^n$.
