We investigate the luminosity functions of Lyman break galaxies (LBGs) at zåisebox-0.5ex~4 and z\i̊sebox-0.5ex~5 based on optical imaging data obtained in the Subaru Deep Field project. Three samples of LBGs in a contiguous area of 875 arcmin$^2$ are constructed. One consists of 3808 LBGs at z\rs̊ebox-0.5ex~4 down to i'=26.85 selected with the B-R versus R-i' diagram. The other two consist of 539 and 240 LBGs at z\rae̊box-0.5ex~5 down to z'=26.05 selected with two kinds of two-color diagram: V-i' versus i'-z' and R-i' versus i'-z'. The adopted selection criteria are proved to be fairly reliable by spectroscopic observations. We derive the luminosity functions of the LBGs at rest-frame ultraviolet wavelengths down to M$_UV$=-19.2 at z\raib̊ox-0.5ex~4 and M$_UV$=-20.3 at z\raiso̊x-0.5ex~5. We find clear evolution of the luminosity function over the redshift range 0<=z<=6, which is accounted for solely by a change in the characteristic magnitude M$^*$. We examine the evolution of the cosmic star formation rate (SFR) density and its luminosity dependence over 0<=z<\raisex̊-0.5ex~6. The SFR density contributed from brighter galaxies is found to change more drastically with cosmic time. The contribution from brighter galaxies has a sharp peak around z=3-4, while that from fainter galaxies evolves relatively mildly with a broad peak at earlier epochs. Combining the observed SFR density with the standard cold dark matter model, we compute the cosmic SFR per unit baryon mass in dark halos, that is, the specific SFR. The specific SFR is found to scale with redshift as (1+z)$^3$ up to z\raiseb̊-0.5ex~4, implying that the efficiency of star formation is on average higher at higher redshift in proportion to the cooling rate within dark halos, while this is not simply the case at z>\raisebo-̊0.5ex~4. Based on data collected at the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.