Caging grasps restrict object motion without requiring complete immobilization, providing a robust alternative to force- and formclosure grasps. Energy-bounded cages are a new class of caging grasps that relax the requirement of complete caging in the presence of external forces such as gravity. In this paper, we address the problem of synthesizing energy-bounded cages by identifying optimal gripper and force-direction configurations that require the largest increases in potential energy for the object to escape. We present Energy-Bounded-Cage- Synthesis-2D (EBCS-2D), a sampling-based algorithm that uses persistent homology, a recently-developed multiscale approach for topological analysis, to efficiently compute candidate rigid configurations of obstacles that form energy-bounded cages of an object from an α-shape approximation to the configuration space. We also show that constant velocity pushing in the horizontal plane generates an energy field analogous to gravity in the vertical plane that can be analyzed with our algorithm. EBCS-2D runs in O(s 3 + sn 2 ) time where s is the number of samples and n is the total number of object and obstacle vertices, where typically n << s. We observe runtimes closer to O(s) for fixed n. We implement EBCS-2D using the Persistent Homology Algorithms Toolbox (PHAT) and study performance on a set of seven planar objects and four gripper types. Experiments suggest that EBCS-2D takes 2-3 minutes on a 6 core processor with 200,000 pose samples. We also confirm that an RRT* motion planner is unable to find escape paths with lower energy. Physical experiments suggest that push grasps synthesized by EBCS-2D are robust to perturbations. Additional proofs, data, and code are available at http://berkeleyautomation.github.io/caging/.