With the advent of the European Extremely Large Telescope and its high resolution spectrograph HIRES, the era of the so called “on-the-fly cosmology” will soon become true. In this talk I will review how the E-ELT-HIRES, in which Italy and INAF-OABr will be deeply involved, allows to watch the Universe expanding in less than 20 years of observations from commissioning. Thanks to its extreme stability and the superior signal-to-noise ratio reached throughout its aperture, astronomer will be able to performe a full Sandage Test for the first time recovering the dynamics of the Universe at various redshift. Precise measurement of redshift drift will allow to put constraint on equation of state of dark energy and on densities of energies that dominate the dynamics of the Universe. Moreover, also fundamental physics will take advantage of this exciting instrumentation through the precise measurement of variation of fine-strucuture constant along cosmic time and the even more (and more) accurate observation of chemical abundances of metal poor stars to further constrain the BBN.