Broadband beamformers with constant mainlobe response over the frequency of interest are desirable in many applications including underwater acoustics, ultrasonics, acoustic imaging and communications, and so on. Solutions to this problem have been presented for specific array geometry often requiring a larger number of sensors. And the array pattern synthesis error minimization is employed for the whole field of view, which leads to suboptimal designs. In this paper, a broadband array pattern synthesis approach to designing time-domain constant mainlobe response beamformer is proposed. By imposing constraints both on the mainlobe spatial response variation over frequency and on the sidelobes of the beamformer, several optimization criteria are presented and the corresponding convex second-order cone programming implementations are given. In this approach, no preliminary desired beampattern is required and the beam responses variation minimization is employed only in the mainlobe region and not in the sidelobe regions, which improves the beamformer mainlobe synthesis accuracy. Equally, one can obtain lower sidelobes at the same mainlobe synthesis accuracy. This approach is applicable to arrays with arbitrary geometry. Simulation and experimental results are presented to illustrate the effectiveness of this approach. Performance comparisons of the proposed beamformers and the existing beamformer are also provided.