  Certain scattering amplitudes in the gravitational sector of type II string
theory on K3 x T^2 are found to be computed by correlation functions of the N=4
topological string. This analysis extends the already known results for K3 by
Berkovits and Vafa, which correspond to six-dimensional terms in the effective
action, involving four Riemann tensors and 4g-4 graviphotons, R^4T^{4g-4}, at
genus g. We find two additional classes of topological amplitudes that use the
full internal SCFT of K3 x T^2. One of these string amplitudes is mapped to a
1-loop contribution in the heterotic theory, and is studied explicitly. It
corresponds to the four-dimensional term R^2(dd\Phi)^2T^{2g-4}, with \Phi a
Kaluza-Klein graviscalar from T^2. Finally, the generalization of the
harmonicity relation for its moduli dependent coupling coefficient is obtained
and shown to contain an anomaly, generalizing the holomorphic anomaly of the
N=2 topological partition function F_g.
