We present the first systematic study of the molecular gas and star formation efficiency in a sample of ten narrow-line Seyfert 1 galaxies selected to have X-ray Ultra Fast Outflows and, therefore, to potentially show AGN feedback effects. CO observations were obtained with the IRAM 30-m telescope in six galaxies and from the literature for four galaxies. We derived the stellar mass, star formation rate, AGN, and FIR dust luminosities by fitting the multi-band spectral energy distributions with the CIGALE code. Most of the galaxies in our sample lie above the main sequence (MS), and the molecular depletion time is one to two orders of magnitude shorter than the one typically measured in local star-forming galaxies. Moreover, we found a promising correlation between the star formation efficiency and the Eddington ratio, as well as a tentative correlation with the AGN luminosity. The role played by the AGN activity in the regulation of star formation within the host galaxies of our sample remains uncertain (little or no effect? positive feedback?). Nevertheless, we can conclude that quenching by the AGN activity is minor and that star formation will likely stop in a short time due to gas exhaustion by the current starburst episode.
