  We investigate the special K\"ahler geometry of the base of the Hitchin
integrable system in terms of spectral curves and topological recursion. The
Taylor expansion of the special K\"ahler metric about any point in the base may
be computed by integrating the $g = 0$ Eynard-Orantin invariants of the
corresponding spectral curve over cycles. In particular, we show that the
Donagi-Markman cubic is computed by the invariant $W^{(0)}_3$. We use
topological recursion to go one step beyond this and compute the symmetric
quartic of second derivatives of the period matrix.
