We present new measurements of the luminosity function (LF) of Luminous Red Galaxies (LRGs) from the Sloan Digital Sky Survey (SDSS) and the 2dF-SDSS LRG and Quasar (2SLAQ) survey. We have carefully quantified, and corrected for, uncertainties in the K and evolutionary corrections, differences in the colour selection methods, and the effects of photometric errors, thus ensuring we are studying the same galaxy population in both surveys. Using a limited subset of 6326 SDSS LRGs (with 0.17<z<0.24) and 1725 2SLAQ LRGs (with 0.5 <z<0.6), for which the matching colour selection is most reliable, we find no evidence for any additional evolution in the LRG LF, over this redshift range, beyond that expected from a simple passive evolution model. This lack of additional evolution is quantified using the comoving luminosity density of SDSS and 2SLAQ LRGs, brighter than M_r - 5logh = -22.5, which are 2.51+/-0.03 x 10^-7 L_sun Mpc^-3 and 2.44+/-0.15 x 10^-7 L_sun Mpc^-3 respectively (<10% uncertainty). We compare our LFs to the COMBO-17 data and find excellent agreement over the same redshift range. Together, these surveys show no evidence for additional evolution (beyond passive) in the LF of LRGs brighter than M_r - 5logh = -21 (or brighter than L*). We test our SDSS and 2SLAQ LFs against a simple ``dry merger model for the evolution of massive red galaxies and find that at least half of the LRGs at z=0.2 must already have been well-assembled (with more than half their stellar mass) by z=0.6. This limit is barely consistent with recent results from semi-analytical models of galaxy evolution.
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