  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. We 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, we assess the impact of the
cross calibration uncertainties on cosmology. Using the HIFLUGCS sample,
consisting of the 64 X-ray brightest galaxy clusters, we constrain the ICM
temperatures through spectral fitting in the same, mostly isothermal, regions
and compare them. Our work is an extension to a previous one using X-ray
clusters by the IACHEC. Performing spectral fitting in the full energy band we
find that best-fit temperatures determined with XMM-Newton/EPIC are
significantly lower than Chandra/ACIS temperatures. We demonstrate that effects
like multitemperature structure and different relative sensitivities of the
instruments at certain energy bands cannot explain the observed differences. We
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. Such a
shift is insufficient to significantly alleviate the tension between Planck CMB
anisotropies and SZ plus XMM-Newton cosmological constraints.
