We compute the L-2-Betti numbers of the free C*-tensor categories, which are the representation categories of the universal unitary quantum groups A(u)(F). We show that the L-2-Betti numbers of the dual of a compact quantum group G are equal to the L-2-Betti numbers of the representation category Rep. (G) and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of L-2-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first L-2-Betti number in terms of a generating set of a C*-tensor category.