In the 80's H. Masur and W. Veech defined two numerical in-variants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymptotic behavior of quasi-modular forms, D. Chen, M. Moeller, and D. Zagier proved that this conjecture holds for strata of differentials with simple zeros. Here, with a mild assumption of existence of a good metric, we show that the conjecture holds for the other extreme case, i.e. for strata of differentials with a unique zero. Our main ingredient is the expression of the numerical invariants of these strata in terms of Hodge integrals on moduli spaces of curves.