Cosmological constraints from clusters rely on accurate gravitational mass estimates, which strongly depend on cluster gas temperature measurements. Therefore, systematic calibration differences may result in biased, instrument-dependent cosmological constraints. This is of special interest in the light of the tension between the Planck results of the primary temperature anisotropies of the CMB and Sunyaev-Zel'dovich plus X-ray cluster counts analyses. I quantify in detail the systematics and uncertainties of the cross-calibration of the effective area between five X-ray instruments, EPIC-MOS1/MOS2/PN onboard XMM-Newton and ACIS-I/S onboard Chandra, and the influence on temperature measurements. Furthermore, I assess the impact of the cross calibration uncertainties on cosmology. Using the HIFLUGCS sample, consisting of the 64 X-ray brightest galaxy clusters, I constrain the ICM temperatures through spectral fitting in the same, mostly isothermal, regions and compare them. Performing spectral fitting in the full energy band I find that best-fit temperatures determined with XMM-Newton/EPIC are significantly lower than Chandra/ACIS temperatures. I demonstrate that effects like multitemperature structure cannot explain the observed differences. I conclude that using XMM-Newton/EPIC, instead of Chandra/ACIS to derive full energy band temperature profiles for cluster mass determination results in an 8% shift towards lower OmegaM values and <1% shift towards higher sigma8 values in a cosmological analysis of a complete sample of galaxy clusters. For this conclusion, the Tinker+08 halo mass function was used, which is a sensitive probe of cosmological parameters. Such a shift is insufficient to significantly alleviate the tension between Planck CMB anisotropies and SZ plus XMM-Newton cosmological constraints. I will close by giving an outlook on what difficulties occur when deriving hydrostatic masses and constraining cosmological parameters from a complete sample.