Antenna arrays based on the space-filling Peano-Gosper (PG) curve possess several beneficial properties, including low sidelobes, a 2:1 bandwidth with suppressed grating lobes, and a convenient modular architecture. Due to their triangular lattice element distribution, these arrays are limited in bandwidth up to a minimum element spacing of approximately one-wavelength. One way to extend the bandwidth of these arrays is by introducing perturbations into the basic array geometry to break up the regular triangular distribution of elements. With the proper perturbation scheme, the overall geometry of the arrays can be greatly varied using a relatively small number of design parameters while retaining the modularity of the initial array. Recently, generalized forms of the Peano-Gosper curve have been discovered. These curves possess varying degrees of modularity that are not inherent to the conventional PG curve. Similar to conventional PG arrays, arrays based on generalized PG curves have a limited bandwidth due to their triangular lattice element distribution. In this paper, we will extend the application of the basic perturbation technique to generalized PG arrays with the goal of appreciably extending their bandwidth. The maximum bandwidth enhancement that is obtained via the perturbed generalized PG curves is significantly greater than that of the perturbed conventional PG curves. The design of these arrays will be carried out by a hybrid GA/Nelder-Mead optimizer. A design example with a 6.9:1 bandwidth will be shown to demonstrate the efficacy of this technique.