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We introduce a new test to study the Cosmological Principle with galaxyclusters. Galaxy clusters exhibit a tight correlation between the luminosityand temperature of the X-ray-emitting intracluster medium. While the luminositymeasurement depends on cosmological parameters through the luminosity distance,the temperature determination is cosmology-independent. We exploit thisproperty to test the isotropy of the luminosity distance over the fullextragalactic sky, through the normalization $a$ of the $L_X-T$ scalingrelation and the cosmological parameters $\Omega_m$ and $H_0$. We use twoalmost independent galaxy cluster samples: the ASCA Cluster Catalog (ACC) andthe XMM Cluster Survey (XCS-DR1). Interestingly enough, these two samplesappear to have the same pattern for $a$ with respect to the Galactic longitude.We also identify one sky region within $l\sim (-15^o,90^o)$ (Group A) thatshares very different best-fit values for $a$ for both samples. We find thedeviation of Group A to be $2.7\sigma$ for ACC and $3.1\sigma$ for XCS-DR1.This tension is not relieved after excluding possible outliers or after aredshift conversion to the CMB frame is applied. Using also the HIFLUGCSsample, we show that a possible excess of cool-core clusters in this region,cannot explain the obtained deviations. Moreover, we tested for a dependence ofthe $L_X-T$ relation on supercluster environment. We indeed find a trend forsupercluster members to be underluminous compared to field clusters. However,the fraction of supercluster members is similar in the different sky regions.Constraining $\Omega_m$ and $H_0$ via the redshift evolution of $L_X-T$ and theluminosity distance, we obtain approximately the same deviation amplitudes asfor $a$. The observed behavior of $\Omega_m$ for the sky regions that coincidewith the CMB dipole is similar to what was found with other cosmological probesas well.

Anisotropy of the galaxy cluster X-ray luminosity-temperature relation