  We study the impact of relatively weak AGN feedback on the interstellar
medium of intermediate and massive elliptical galaxies. We find that the AGN
activity, while globally heating the ISM, naturally stimulates some degree of
hot gas cooling on scales of several kpc. This process generates the persistent
presence of a cold ISM phase, with mass ranging between 10$^4$ and $\gtrsim$ 5
$\times$ 10$^7$ M$_\odot$, where the latter value is appropriate for group
centered, massive galaxies. Widespread cooling occurs where the ratio of
cooling to free-fall time before the activation of the AGN feedback satisfies
$t_{cool}/t_{ff} \lesssim 70$, that is we find a less restrictive threshold
than commonly quoted in the literature. This process helps explaining the body
of observations of cold gas (both ionized and neutral/molecular) in Ellipticals
and, perhaps, the residual star formation detected in many early-type galaxies.
The amount and distribution of the off-center cold gas vary irregularly with
time. The cold ISM velocity field is irregular, initially sharing the
(outflowing) turbulent hot gas motion. Typical velocity dispersions of the cold
gas lie in the range 100-200 km/s. Freshly generated cold gas often forms a
cold outflow and can appear kinematically misaligned with respect to the stars.
We also follow the dust evolution in the hot and cold gas. We find that the
internally generated cold ISM has a very low dust content, with representative
values of the dust-to-gas ratio of 10$^{-4}$- 10$^{-5}$. Therefore, this cold
gas can escape detection in the traditional dust-absorption maps.
