In this paper, we study the behaviour of the entropy function of higher-dimensional shifts of finite type. We construct a topologically mixing $\mathbb{Z}^2$ shift of finite type whose ergodic invariant measures are connected in the $\overline{d}$ topology and whose entropy function has a strictly local maximum. We also construct a topologically mixing $\mathbb{Z}^2$ shift of finite type X with the property that there is a uniform gap between the topological entropy of X and the topological entropy of any subshift of X with stronger mixing properties. Our examples illustrate the necessity of strong topological mixing hypotheses in existing higher-dimensional representation and embedding theorems.