Large spectroscopic surveys devoted to the study of the Milky Way, including Gaia, use automated pipelines to determine the atmospheric parameters of millions of stars. The Gaia FGK benchmark stars are reference stars with Teff and logg derived through fundamental relations, independently of spectroscopy, to be used as anchors for the parameter scale. The first and second versions of the sample have been extensively used for that purpose, and more generally to help constrain stellar models. We provide the third version of the Gaia FGK benchmark stars, an extended set intended to improve the calibration of spectroscopic surveys, and their interconnection. We have compiled about 200 candidates that have precise measurements of angular diameters and parallaxes. We determined their bolometric fluxes by fitting their spectral energy distribution. Masses were determined using two sets of stellar evolution models. In a companion paper, we describe the determination of metallicities and detailed abundances. We provide a new set of 192 Gaia FGK benchmark stars with their fundamental Teff and logg, and with uncertainties lower than 2% for most stars. Compared to the previous versions, the homogeneity and accuracy of the fundamental parameters are significantly improved thanks to the high quality of the Gaia photometric and astrometric data.