We perform forecasts for the Hubble parameter H_0 and for the parameter Xi_0 that describes modified gravitational-wave propagation, using information from the binary neutron star (BNS) mass function, for Einstein Telescope (ET), taken either in the triangle or in the ``2L'' configuration, as well as for detector network made by ET together with a 40-km Cosmic Explorer (CE). We restrict ourselves to BNSs with a large signal-to-noise ratio, SNR>50, which still give O(10^3) events yr-1, and we perform a full joint cosmology-population Bayesian inference. We find that, for ET in isolation, the two ET configurations perform comparably, yielding uncertainties of 12% and 11% on H_0 for the triangular and 2L designs, respectively, and 18% uncertainty on Xi_0 in both cases. For networks including ET and CE, we can constrain H_0 and Xi_0 to precisions of 9% and 6%, respectively. These results should be taken as a very conservative estimate of third-generation detectors' capabilities as a consequence of the high SNR cut. We project the constraints on the Lambda CDM expansion history and find that ET alone (triangular and 2L configurations) achieves its best precision on H(z) at z=0.23 and z=0.28, yielding a 10% and 6% precision, respectively. When CE is added to the network, the precision improves to 4% and 3% at z=0.37 and z=0.38, respectively.
