We present a photometric variability survey of young planetary-mass objects using the New Technology Telescope in the J_S and K_S bands. Surface gravity plays an important role in the atmospheric structure of brown dwarfs, as young low-gravity L dwarfs have a higher variability rate than field L dwarfs. In this study, we extend variability studies to young T-type planetary-mass objects and investigate the effects of surface gravity on the variability of L and T dwarfs across a large sample. We conduct continuous monitoring for 18 objects with spectral types from L5 to T8 and detect four new variables and two variable candidates. Combining with previous variability surveys of field and young L and T objects, we find that young objects tend to be more variable than field objects within peak-to-peak variability amplitude ranges of 0.5-10 per cent and period ranges of 1.5-20 h. For the first time, we constrain the variability rate of young T dwarfs to be $56_{-18}^{+20}$ per cent compared to $25_{-7}^{+8}$ per cent for field T dwarfs. Both field and young samples have higher variability rates at the L/T transition than outside the L/T transition. The differences in the variability rates between field and young samples are about 1σ and therefore larger sample sizes are needed to confirm and refine the results. Besides the L/T transition, young L dwarfs with strong variability tend to assemble in a narrow spectral type range of L6-L7.5. This work supports the critical role of surface gravity on the atmospheric structure from L to T spectral types.