We measure the star formation rate (SFR) as a function of redshift z up to z ~4.5, based on B, I and (I+B) selected galaxy catalogues from the FORS Deep Field (FDF) and the K-selected catalogue from the GOODS-South field. Distances are computed from spectroscopically calibrated photometric redshifts accurate to (Delta_z / (z_spec+1)) ~0.03 for the FDF and ~0.056 for the GOODS-South field. The SFRs are derived from the luminosities at 1500 Angstroem. We find that the total SFR estimates derived from B, I and I+B catalogues agree very well ($lsim 0.1$ dex) while the SFR from the K catalogue is lower by ~0.2 dex. We show that the latter is solely due to the lower star-forming activity of K-selected intermediate and low luminosity (L<L_*) galaxies. The SFR of bright (L>L_*) galaxies is independent of the selection band, i.e. the same for B, I, (I+B), and K-selected galaxy samples. At all redshifts, luminous galaxies (L>L_*) contribute only ~1/3 to the total SFR. There is no evidence for significant cosmic variance between the SFRs in the FDF and GOODs-South field, ~0.1 dex, consistent with theoretical expectations. The SFRs derived here are in excellent agreement with previous measurements provided we assume the same faint-end slope of the luminosity function as previous works (alpha ~ -1.6). However, our deep FDF data indicate a shallower slope of alpha=-1.07, implying a SFR lower by ~0.3 dex. We find the SFR to be roughly constant up to z ~4 and then to decline slowly beyond, if dust extinctions are assumed to be constant with redshift.
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