  The presence of dilute hot cavities in the intracluster medium (ICM) at the
cores of clusters of galaxies changes the relation between gas temperature and
its X-ray emission properties. Using the hydrostatic equations of a porous
medium we solve for the ICM density for a given temperature as a function of
the filling factor of dilute bubbles. We find that at a given temperature, the
core X-ray luminosity increases with the filling factor. If the frequency of
AGNs in clusters were higher in the past, then the filling factor could
correspondingly be significant, with implications for the cluster scaling
relations at high redshifts. This is especially important for the core
properties, including the temperature-luminosity ($L_X-T$) relation and
estimates of the core gas mass. The results imply an epoch-dependent
sensitivity of the $L_X-T$ relation in the core to the porosity of the ICM.
Detection of such an effect would give new insights into AGN feedback.
