A coronagraphic starlight suppression system situated on a future flagship space observatory offers a promising avenue to image Earth-like exoplanets and search for biomarkers in their atmospheric spectra. One NASA mission concept that could serve as the platform to realize this scientific breakthrough is the Large UV/Optical/IR Surveyor (LUVOIR). Such a mission would also address a broad range of topics in astrophysics with a multiwavelength suite of instruments. The apodized pupil Lyot coronagraph (APLC) is one of several coronagraph design families that the community is assessing as part of NASAs Exoplanet Exploration Program Segmented aperture coronagraph design and analysis (SCDA) team. The APLC is a Lyot-style coronagraph that suppresses starlight through a series of amplitude operations on the on-axis field. Given a suite of seven plausible segmented telescope apertures, we have developed an object-oriented software toolkit to automate the exploration of thousands of APLC design parameter combinations. This has enabled us to empirically establish relationships between planet throughput and telescope aperture geometry, inner working angle, bandwidth, and contrast level. In parallel with the parameter space exploration, we have investigated several strategies to improve the robustness of APLC designs to fabrication and alignment errors. We also investigate the combination of APLC with wavefront control or complex focal plane masks to improve inner working angle and throughput. Preliminary scientific yield evaluations based on design reference mission simulations indicate the APLC is a very competitive concept for surveying the local exoEarth population with a mission like LUVOIR.