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The environment around protoplanetary disks (PPDs) regulates processes whichdrive the chemical and structural evolution of circumstellar material. Weperform a detailed empirical survey of warm molecular hydrogen (H$_2$)absorption observed against H I-Ly$\alpha$ (Ly$\alpha$: $\lambda$ 1215.67{\AA}) emission profiles for 22 PPDs, using archival Hubble Space Telescope(HST) ultraviolet (UV) spectra from the Cosmic Origins Spectrograph (COS) andthe Space Telescope Imaging Spectrograph (STIS) to identify H$_2$ absorptionsignatures and quantify the column densities of H$_2$ ground states in eachsightline. 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$_{exc}$ $\gtrsim$ 20,000 K)away from thermal equilibrium populations (T(H$_2$) $\gtrsim$ 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 targetswith detectable C IV-pumped H$_2$ fluorescence. We compare N(H$_2$) andN(H$_2$)$_{nLTE}$ to circumstellar observables and find that N(H$_2$)$_{nLTE}$correlates with X-ray and FUV luminosities, but no correlations are observedwith 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 theH$_2$ fluorescence observed in PPDs, so we speculate that this H$_2$ mayinstead be associated with a diffuse, hot, atomic halo surrounding theplanet-forming disk. We create a simple photon-pumping model for each target totest this hypothesis and find that Ly$\alpha$ efficiently pumps H$_2$ levelswith T$_{exc}$ $\geq$ 10,000 K out of thermal equilibrium.

Signatures of Hot Molecular Hydrogen Absorption from Protoplanetary Disks: I. Non-thermal Populations