We consider the $n$-dimensional random temporal hypercube, i.e., the $n$-dimensional hypercube graph with its edges endowed with i.i.d. continuous random weights. We say that a vertex $w$ is accessible from another vertex $v$ if and only if there is a path from $v$ to $w$ with increasing edge weights. We study accessible "direct" paths from a fixed vertex to its antipodal point and show that as $n \to \infty$, the number of such paths converges in distribution to a mixed Poisson law with mixture given by the product of two independent exponentials with rate $1$. Our proof makes use of the Chen-Stein method, coupling arguments, as well as combinatorial arguments which show that typical pairs of accessible paths have small overlap.