The nature and origin of the cold interstellar medium (ISM) in early-type galaxies are still a matter of debate, and understanding the role of this component in galaxy evolution and in fuelling the central supermassive black holes requires more observational constraints. Here, we present a multiwavelength study of the ISM in eight nearby, X-ray and optically bright, giant elliptical galaxies, all central dominant members of relatively low-mass groups. Using far-infrared spectral imaging with the Herschel Photodetector Array Camera & Spectrometer, we map the emission of cold gas in the cooling lines of [C II]λ157 μm, [O I] λ63 μm and [O Ib] λ145 μm. Additionally, we present Hα [N II] imaging of warm ionized gas with the Southern Astrophysical Research (SOAR) telescope, and a study of the thermodynamic structure of the hot X-ray emitting plasma with Chandra. All systems with extended Hα emission in our sample (6/8 galaxies) display significant [C II] line emission indicating the presence of reservoirs of cold gas. This emission is cospatial with the optical Hα [N II] emitting nebulae and the lowest entropy soft X-ray emitting plasma. The entropy profiles of the hot galactic atmospheres show a clear dichotomy, with the systems displaying extended emission-line nebulae having lower entropies beyond r ≳ 1 kpc than the cold-gas-poor systems. We show that while the hot atmospheres of the cold-gas-poor galaxies are thermally stable outside of their innermost cores, the atmospheres of the cold-gas-rich systems are prone to cooling instabilities. This provides considerable weight to the argument that cold gas in giant ellipticals is produced chiefly by cooling from the hot phase. We show that cooling instabilities may develop more easily in rotating systems and discuss an alternative condition for thermal instability for this case. The hot atmospheres of cold-gas-rich galaxies display disturbed morphologies indicating that the accretion of clumpy multiphase gas in these systems may result in variable power output of the AGN jets, potentially triggering sporadic, larger outbursts. In the two cold-gas-poor, X-ray morphologically relaxed galaxies of our sample, NGC 1399 and NGC 4472, powerful AGN outbursts may have destroyed or removed most of the cold gas from the cores, allowing the jets to propagate and deposit most of their energy further out, increasing the entropy of the hot galactic atmospheres and leaving their cores relatively undisturbed.
