Let $B$ be a finite CW complex and $G$ a compact connected Lie group. We show
that the number of gauge groups of principal $G$-bundles over $B$ is finite up
to $A_n$-equivalence for $n<\infty$. As an example, we give a lower bound of
the number of $A_n$-equivalence types of gauge groups of principal
$\SU(2)$-bundles over $S^4$.