What I can find on the subject are the papers by Fricke and the Habermanns: http://www.researchgate.net/publication/227336993_On_the_geometry_of_moduli_spaces_of_symplectic_structures and http://www.sciencedirect.com/science/article/pii/S0926224505000197
They study the connected components of this moduli space and construct a pseuo-Riemannian metric on it of predictable signature.
I'm trying to get a sense of this moduli space and some current problems in the field, but it seems like a subject that isn't well-studied, and perhaps there aren't many interesting examples to explore?