We present here the cosmo-SLICS, a new suite of simulations specially designed for the analysis of current and upcoming weak lensing data beyond the standard two-point cosmic shear. We sample the [Ω m , σ 8 , h, w 0 ] parameter space at 25 points organised in a Latin hyper-cube, spanning a range that contains most of the 2σ posterior distribution from ongoing lensing surveys. At each of these nodes we evolve a pair of N-body simulations in which the sampling variance is highly suppressed, and ray-trace the volumes 800 times to further increase the effective sky coverage. We extract a lensing covariance matrix from these pseudo-independent lightcones and show that it closely matches a brute-force construction based on an ensemble of 800 truly independent N-body runs. More precisely, a Fisher analysis reveals that both methods yield marginalized two-dimensional constraints that vary by less than 6% in area, a result that holds under different survey specifications and that matches to within 15% the area obtained from an analytical covariance calculation. Extending this comparison with our 25 wCDM models, we probe the cosmology dependence of the lensing covariance directly from numerical simulations, reproducing remarkably well the Fisher results from the analytical models at most cosmologies. We demonstrate that varying the cosmology at which the covariance matrix is evaluated in the first place might have an order of magnitude greater impact on the parameter constraints than varying the choice of covariance estimation technique. We present a test case in which we generate fast predictions for both the lensing signal and its associated variance with a flexible Gaussian process regression emulator, achieving an accuracy of a few percent on the former and 10% on the latter.