The physics of diffuse gas accretion and the properties of the cold and hot modes of accretion on to proto-galaxies between z= 2 and 5.4 is investigated using the large cosmological simulation performed with the ramses code on the MareNostrum supercomputing facility. Galactic winds, chemical enrichment, ultraviolet background heating and radiative cooling are taken into account in this very high resolution simulation. Using accretion-weighted temperature histograms, we have performed two different measurements of the thermal state of the gas accreted towards the central galaxy. The first measurement, performed using accretion-weighted histograms on a spherical surface of radius 0.2Rvir centred on the densest gas structure near the halo centre of mass, is a good indicator of the presence of an accretion shock in the vicinity of the galactic disc. We define the hot shock mass, Mshock, as the typical halo mass separating cold dominated from hot dominated accretion in the vicinity of the galaxy. The second measurement is performed by radially averaging histograms between 0.2Rvir and Rvir, in order to detect radially extended structures such as gas filaments: this is a good proxy for detecting cold streams feeding the central galaxy. We define Mstream as the transition mass separating cold dominated from hot dominated accretion in the outer halo, marking the disappearance of these cold streams. We find a hot shock transition mass of Mshock= 1011.6M⊙ (dark matter), with no significant evolution with redshift. Conversely, we find that Mstream increases sharply with z. Our measurements are in agreement with the analytical predictions of Birnboim & Dekel and Dekel & Birnboim, if we correct their model by assuming low metallicity(≤10−3Z⊙)for the filaments, correspondingly to our measurements. Metal enrichment of the intergalactic medium is therefore a key ingredient in determining the transition mass from cold to hot dominated diffuse gas accretion. We find that the diffuse cold gas supply at the inner halo stops at z= 2 for objects with stellar masses of about 1011.1M⊙, which is close to the quenching mass determined observationally by Bundy et al. However, its evolution with z is not well constrained, making it difficult to rule out or confirm the need for an additional feedback process such as active galactic nuclei.