This study investigates hydrostatic blockage and excitation of gravity waves in large offshore wind farms within convective boundary layers (CBLs), a regime currently underexplored compared to stable and neutral conditions. Using large-eddy simulations, the performance of a 1.5 GW wind farm is analysed across eight scenarios representative of cold-air outbreaks. These cases vary in capping-inversion height (<em>H</em> = {480,980} m) and surface temperature difference (Δ<em>θ</em> = {0,3,6,9} K), producing various convective structures, including rolls and cells. The results demonstrate that intensified convective mixing and boundary layer growth, driven by higher Δ<em>θ</em>, significantly alter flow physics. Enhanced buoyancy-driven turbulence improves wake recovery, leading to more uniform power distribution. Simultaneously, the growth of the boundary layer slightly reduces the magnitude of both unfavourable and favourable pressure gradients, an effect particularly pronounced in shallow boundary layers. Efficiency analysis reveals that while wake efficiency (<em>η</em><sub>w</sub>) increases with convective intensity, whereas non-local efficiency (<em>η</em><sub>nl</sub>) remains largely unaffected. Consequently, overall farm efficiency (<em>η</em><sub>f</sub>) improves under stronger convective forcing, though sensitivity diminishes at higher Δ<em>θ</em>. Despite increased mixing, hydrostatic blockage and gravity waves persist; even in deep boundary layers, a non-local efficiency of approximately 90 % was recorded. Decreasing the boundary layer height, amplifies the influence of both unfavourable and favourable pressure gradients on wind farm efficiency, whereas the relative contribution of convective mixing becomes less significant.