Context. Classical Cepheids are primary distance indicators and a crucial stepping stone in determining the present-day value of the Hubble constant H-0 to the precision and accuracy required to constrain apparent deviations from the ACDM Concordance Cosmological Model. Aims. We measured the iron and oxygen abundances of a statistically significant sample of 89 Cepheids in the Large Magellanic Cloud (LMC), one of the anchors of the local distance scale, quadrupling the prior sample and including 68 of the 70 Cepheids used to constrain H-0 by the SHOES program. The goal is to constrain the extent to which the luminosity of Cepheids is influenced by their chemical composition, which is an important contributor to the uncertainty on the determination of the Hubble constant itself and a critical factor in the internal consistency of the distance ladder. Methods. We derived stellar parameters and chemical abundances from a self-consistent spectroscopic analysis based on equivalent width of absorption lines. Results. The iron distribution of Cepheids in the LMC can be very accurately described by a single Gaussian with a mean [Fe/H] = -0.409 +/- 0.003 dex and sigma = 0.076 +/- 0.003 dex. We estimate a systematic uncertainty on the absolute mean values of 0.1 dex. The width of the distribution is fully compatible with the measurement error and supports the low dispersion of 0.069 mag seen in the near-infrared Hubble Space Telescope LMC period-luminosity relation. The uniformity of the abundance has the important consequence that the LMC Cepheids alone cannot provide any meaningful constraint on the dependence of the Cepheid period-luminosity relation on chemical composition at any wavelength. This revises a prior claim based on a small sample of 22 LMC Cepheids that there was little dependence (or uncertainty) between composition and near-infrared luminosity, a conclusion which would produce an apparent conflict between anchors of the distance ladder with different mean abundance. The chemical homogeneity of the LMC Cepheid population makes it an ideal environment in which to calibrate the metallicity dependence between the more metal-poor Small Magellanic Cloud and metal-rich Milky Way and NGC 4258.