The merger of two compact objects of which at least one is a neutron star is signalled by transient electromagnetic emission in a kilonova (KN). This event is accompanied by gravitational waves and possibly other radiation messengers such as neutrinos or cosmic rays. The electromagnetic emission arises from the radioactive decay of heavy r-process elements synthesized in the material ejected during and after the merger. In this paper we show that the analysis of KNe light curves can provide cosmological distance measurements and constrain the properties of the ejecta. In this respect, MAAT, the new Integral Field Unit in the OSIRIS spectrograph on the 10.4m Gran Telescopio CANARIAS (GTC), is well suited for the study of KNe by performing absolute spectro-photometry over the entire 3600 10 000 angstrom spectral range. Here, we study the most representative cases regarding the scientific interest of KNe from binary neutron stars, and we evaluate the observational prospects and performance of MAAT on the GTC to do the following: (a) study the impact of the equation of state on the KN light curve, and determine to what extent bounds on neutron star (NS) radii or compactness deriving from KN peak magnitudes can be identified and (b) measure the Hubble constant, H-0, with precision improved by up to 40%, when both gravitational wave data and photometric-light curves are used. In this context we discuss how the equation of state, the viewing angle, and the distance affect the precision and estimated value of H-0.