The impact of active galactic nuclei (AGNs) on star formation has implications for our understanding of the relationships between supermassive black holes and their galaxies, as well as for the growth of galaxies over the history of the universe. We report on a high-resolution multiphase study of the nuclear environment in the nearby Seyfert galaxy NGC 2110 using the Atacama Large Millimeter/submillimeter Array, Hubble and Spitzer Space Telescopes, and the Very Large Telescope/SINFONI. We identify a region that is markedly weak in low-excitation CO emission from cold molecular gas, but appears to be filled with ionized and warm molecular gas, which indicates that the AGN is directly influencing the properties of the molecular material. Using multiple molecular gas tracers, we demonstrate that, despite the lack of CO line emission, the surface densities and kinematics of molecular gas vary smoothly across the region. Our results demonstrate that the influence of an AGN on star-forming gas can be quite localized. In contrast to widely held theoretical expectations, we find that molecular gas remains resilient to the glare of energetic AGN feedback.