The $\mathrm{C}^*$-algebra generated by the left-regular representation of $\mathbb{N}^n$ twisted by a $2$-cocycle is a Toeplitz extension of an $n$-dimensional noncommutative torus, on which each vector $r \in [0,\infty)^n$ determines a one-parameter subgroup of the gauge action. I will report on joint work with Z. Afsar, J. Ramagge, and M. Laca, in which we show that the equilibrium states of the resulting $\mathrm{C}^*$-dynamical system are parametrised by tracial states of another noncommutative torus, arising from the restriction of the cocycle to the vanishing coordinates of $r$. These in turn correspond to probability measures on a classical torus whose dimension depends on the degeneracy index of the restricted cocycle. Our results generalise the phase transition on the Toeplitz noncommutative tori used as building blocks in work of Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims.