The relative curvature energetics of two lipids are tested using thermodynamic integration (TI) on four topologically distinct lipid phases. Simulations use TI to switch between choline headgroup lipids (POPC; that prefers to be flat) and ethanolamine headgroup lipids (POPE; that prefer, for example, the inner monolayer of vesicles). The thermodynamical moving of the lipids between planar, inverse hexagonal (H-II), cubic (Q(II); Pn3m space group), and vesicle topologies reveals differences in material parameters that were previously challenging to access. The methodology allows for predictions of two important lipid material properties: the difference in POPC/POPE monolayer intrinsic curvature (Delta J(0)) and the difference in POPC/POPE monolayer Gaussian curvature modulus (Delta(kappa) over bar (m)), both of which are connected to the energetics of topological variation. Analysis of the TI data indicates that, consistent with previous experiment and simulation, the J(0) of POPE is more negative than POPC (Delta J(0) = -0.018 +/- 0.001 angstrom(-1)). The theoretical framework extracts significant differences in (kappa) over bar (m) of which POPE is less negative than POPC by 2.0 to 4.0 kcal/mol. The range of these values is determined by considering subsets of the simulations, and disagreement between these subsets suggests separate mechanical parameters at very high curvature. Finally, the fit of the TI data to the model indicates that the position of the pivotal plane of curvature is not constant across topologies at high curvature. Overall, the results offer insights into lipid material properties, the limits of a single HC model, and how to test them using simulation.