  We introduce a new test to study the Cosmological Principle with galaxy
clusters. Galaxy clusters exhibit a tight correlation between the luminosity
and temperature of the X-ray-emitting intracluster medium. While the luminosity
measurement depends on cosmological parameters through the luminosity distance,
the temperature determination is cosmology-independent. We exploit this
property to test the isotropy of the luminosity distance over the full
extragalactic sky, through the normalization $a$ of the $L_X-T$ scaling
relation and the cosmological parameters $\Omega_m$ and $H_0$. We use two
almost independent galaxy cluster samples: the ASCA Cluster Catalog (ACC) and
the XMM Cluster Survey (XCS-DR1). Interestingly enough, these two samples
appear 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) that
shares very different best-fit values for $a$ for both samples. We find the
deviation 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 a
redshift conversion to the CMB frame is applied. Using also the HIFLUGCS
sample, we show that a possible excess of cool-core clusters in this region,
cannot explain the obtained deviations. Moreover, we tested for a dependence of
the $L_X-T$ relation on supercluster environment. We indeed find a trend for
supercluster 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 the
luminosity distance, we obtain approximately the same deviation amplitudes as
for $a$. The observed behavior of $\Omega_m$ for the sky regions that coincide
with the CMB dipole is similar to what was found with other cosmological probes
as well.
