We report on high average-power, high-energy picosecond fourth-harmonic generation in LBO. The first stages of a Yb:YAG laser chain operating at 1 kHz repetition rate generate few-picosecond 220 mJ chirped pulses at 1030 nm fundamental wavelength. They are frequency-converted in a cascade of three LBO crystals to generate the second-, third-, and fourth-harmonics at 515 nm, 343 nm and 257 nm respectively. Crystals thicknesses and angular phase-matching detuning were calculated as a function of pulse duration through broadband nonlinear optical numerical simulations. Last crystal is both conduction-cooled on edge and surface-cooled at center through forced-air flow to mitigate heating due to nonlinear absorption in the deep-UV and reduce temperature gradients. Chirped-pulse duration was experimentally adjusted to achieve stable 20% overall conversion efficiency. Near-field beam profiles were continuously recorded at 10 Hz, for all four wavelengths involved, together with corresponding energies, showing no significant beam degradation over 50 hours. Temperatures of the two last crystals were monitored, and will help optimize surface cooling for future power ramping-up.