Future galaxy surveys will provide accurate measurements of the matter power spectrum across an unprecedented range of scales and redshifts. The analysis of these data will require one to accurately model the imprint of non-linearities of the matter density field. In particular, these induce a non-Gaussian contribution to the data covariance that needs to be properly taken into account to realise unbiased cosmological parameter inference analyses. Here, we study the cosmological dependence of the matter power spectrum covariance using a dedicated suite of N-body simulations, the Dark Energy Universe Simulation - Parallel Universe Runs (DEUS-PUR) {\it Cosmo}. These consist of 512 realizations for 10 different cosmologies where we vary the matter density $\Omega_m$, the amplitude of density fluctuations $\sigma_8$, the reduced Hubble parameter $h$ and a constant dark energy equation of state $w$ by approximately $10\%$. We use these data to evaluate the first and second derivatives of the power spectrum covariance with respect to a fiducial $\Lambda$CDM cosmology. We find that the variations can be as large as $150\%$ depending on the scale, redshift and model parameter considered. By performing a Fisher matrix analysis we explore the impact of different choices in modelling the cosmological dependence of the covariance. Our results suggest that fixing the covariance to a fiducial cosmology can significantly affect the recovered parameter errors and that modelling the cosmological dependence of the variance while keeping the correlation coefficient fixed can alleviate the impact of this effect.