We present the H alpha luminosity function (LF) derived from a large sample of Lyman break galaxies at z similar to 4.5 over the GOODS-South and North fields. This study makes use of the new, full-depth Spitzer/IRAC [3.6] and [4.5] imaging from the GOODS Re-ionization Era wide-Area Treasury from the Spitzer program. The H alpha flux is derived from the offset between the continuum flux estimated from the best-fit spectral energy distribution, and the observed photometry in IRAC [3.6]. From these measurements, we build the H alpha LF and study its evolution providing the best constraints of this property at high redshift, where spectroscopy of H alpha is not yet available. Schechter parameterizations of the H alpha LF show a decreasing evolution of phi* with redshift, increasing evolution in L *, and no significant evolution in the faint-end slope at high z. We find that star formation rates (SFRs) derived from H alpha are higher than those derived from the rest-frame UV for low SFR galaxies but the opposite happens for the highest SFRs. This can be explained by lower mass galaxies (also lower SFR) having, on average, rising star formation histories (SFHs), while at the highest masses the SFHs may be declining. The SFR function is steeper, and because of the excess SFR(H alpha) compared to SFR(UV) at low SFRs, the SFR density estimated from H alpha is higher than the previous estimates based on UV luminosities.