Let $X = \mathcal{A}^{\mathbb{Z}^d}$, where $d \geq 1$ and $\mathcal{A}$ is a finite set, equipped with the action of the shift map. For a given continuous potential $φ: \mathcal{A}^{\mathbb{Z}^d} \to \mathbb{R}$ and $β>0$ (``inverse temperature''), there exists a (nonempty) set of equilibrium states $\mathrm{ES}(βφ)$. The potential $φ$ is said to exhibit a ``freezing phase transition'' if $\mathrm{ES}(βφ) = \mathrm{ES}(β'φ)$ for all $β, β' > β_c$, while $\mathrm{ES}(βφ) \neq \mathrm{ES}(β'φ)$ for any $β< β_c < β'$, where $β_c\in (0,\infty)$ is a critical inverse temperature depending on $φ$. In this paper, given any proper subshift $X_0$ of $X$, we explicitly construct a continuous potential $φ: X \to \mathbb{R}$ for which there exists $β_c \in (0,\infty)$ such that $\mathrm{ES}(βφ)$ coincides with the set of measures of maximal entropy on $X_0$ for all $β> β_c$, whereas for all $β< β_c$, $μ(X_0)=0$ for all $μ\in \mathrm{ES}(βφ)$. This phenomenon was previously studied only for $d = 1$ in the context of dynamical systems and for restricted classes of subshifts, with significant motivation stemming from quasicrystal models. Additionally, we prove that under a natural summability condition -- satisfied, for instance, by finite-range potentials or exponentially decaying potentials -- freezing phase transitions are impossible.