The environment around protoplanetary disks (PPDs) regulates processes that drive the chemical and structural evolution of circumstellar material. We perform a detailed empirical survey of warm molecular hydrogen (H-2) absorption observed against H I-Ly alpha (Ly alpha:lambda 1215.67) emission profiles for 22 PPDs, using archival Hubble Space Telescope ultraviolet (UV) spectra to identify H-2 absorption signatures and quantify th e column densities of H-2 ground states in each sightline. We compare thermal equilibrium models of H-2 to the observed H-2 rovibrational level distributions. We find that, for the majority of targets, there is a clear deviation in high-energy states (T-exc greater than or similar to 20,000 K) away from thermal equilibrium populations (T(H-2) greater than or similar to 3500 K). We create a metric to estimate the total column density of non-thermal H-2 (N(H-2)(nLTE)) and find that the total column densities of thermal (N(H-2)) and N(H-2)(nLTE) correlate for transition disks and targets with detectable C IV-pumped H-2 fluorescence. We compare N(H-2) and N(H-2)(nLTE) to circumstellar observables and find that N(H-2)(nLTE) correlates with X-ray and far-UV luminosities, but no correlations are observed with the luminosities of discrete emission features (e.g., Ly alpha, C IV). Additionally, N(H-2) and N(H-2)(nLTE) are too low to account for the H-2 fluorescence observed in PPDs, so we speculate that this H-2 may instead be associated with a diffuse, hot, atomic halo surrounding the planet-forming disk. We create a simple photon-pumping model for each target to test this hypothesis and find that Lya efficiently pumps H-2 levels with T-exc >= 10,000 K out of thermal equilibrium.