Using new APEX and JCMT spectroscopy of the CO 2-1 line, we undertake a controlled study of cold molecular gas in moderately luminous Active Galactic Nuclei (AGN) and inactive galaxies from the Luminous Local AGN with Matched Analogs (LLAMA) survey. We use spatially resolved infrared photometry of the LLAMA galaxies from 2MASS, WISE, IRAS & Herschel, corrected for nuclear emission using multi-component spectral energy distribution (SED) fits, to examine the dust-reprocessed star-formation rates (SFRs), molecular gas fractions and star formation efficiencies (SFEs) over their central 1 - 3 kpc. We find that the gas fractions and central SFEs of both active and inactive galaxies are similar when controlling for host stellar mass and morphology (Hubble type). The equivalent central molecular gas depletion times are consistent with the discs of normal spiral galaxies in the local Universe. Despite energetic arguments that the AGN in LLAMA should be capable of disrupting the observable cold molecular gas in their central environments, our results indicate that nuclear radiation only couples weakly with this phase. We find a mild preference for obscured AGN to contain higher amounts of central molecular gas, which suggests a connection between AGN obscuration and the gaseous environment of the nucleus. Systems with depressed SFEs are not found among the LLAMA AGN. We speculate that the processes that sustain the collapse of molecular gas into dense pre-stellar cores may also be a prerequisite for the inflow of material on to AGN accretion disks.
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