Cats in cubes

Answering a recent question of Patchell and Spiro, we show that when a $d$-dimensional cube of side length $n$ is filled with letters, the word $\mathsf{C}\mathsf{A}\mathsf{T}$ can appear contiguously at most $(3^{d-1}\mathrm{/}2)n^d$ times (allowing diagonals); we also characterize when equality occurs and extend our results to words other than $\mathsf{C}\mathsf{A}\mathsf{T}$.