  We study deformations of cluster algebras with several quantum parameters,
called toroidal cluster algebras, which naturally appear in the study of
Grothendieck rings of representations of quantum affine algebras. In this
context, we construct toroidal Grothendieck rings and we establish these are
flat deformations of Grothendieck rings. We prove that for a family of monoidal
categories $\mathscr{C}_1$ of simply-laced quantum affine algebras
categorifying finite-type cluster algebras, the toroidal Grothendieck ring has
a natural structure of a toroidal cluster algebra.
