  The 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. We show that the equilibrium states
of the resulting C*-dynamical system are parametrised by tracial states of the
noncommutative torus corresponding to 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 a certain degeneracy index of
the restricted cocycle. Our results generalise the phase transition on the
Toeplitz noncommutative tori used as building blocks in recent work of
Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims.
