  In this paper we prove the existence of a pseudo-K\"ahler structure on the
deformation space $\mathcal{B}_0(T^2)$ of properly convex $\mathbb R\mathbb
P^2$-structures over the torus. In particular, the pseudo-Riemannian metric and
the symplectic form are compatible with the complex structure inherited from
the identification of $\mathcal{B}_0(T^2)$ with the complement of the zero
section of the total space of the bundle of cubic holomorphic differentials
over the Teichm\"uller space. We show that the $S^1$-action on
$\mathcal{B}_0(T^2)$, given by rotation of the fibers, is Hamiltonian and it
preserves both the metric and the symplectic form. Finally, we prove the
existence of a moment map for the $\mathrm{SL}(2,\mathbb R)$-action over
$\mathcal{B}_0(T^2)$.
