An $r$-uniform hypergraph $H$ consists of a set of vertices $V$ and a set of edges whose elements are $r$-subsets of $V$. We define a hypertree to be a connected hypergraph which contains no cycles. A hypertree spans a hypergraph $H$ if it is a subhypergraph of $H$ which contains all vertices of $H$. Greenhill, Isaev, Kwan and McKay (2017) gave an asymptotic formula for the average number of spanning trees in graphs with given, sparse degree sequence. We prove an analogous result for $r$-uniform hypergraphs with given degree sequence $\boldsymbol{k} = (k_1,\ldots, k_n)$. Our formula holds when $r^5 k_{\max}^3 = o((kr-k-r)n)$, where $k$ is the average degree and $k_{\max}$ is the maximum degree.