Already shortly after the discovery of the AdS/CFT correspondence it became clear that holography can deal remarkably easily with the finite-temperature physics of the boundary system. The key is that one can account for the thermal physics of the strongly interacting critical state in the boundary by adding a black hole in the deep interior in the AdS bulk. As we will explain in detail in the first section, the temperature of the boundary system turns out to be identical to the Hawking temperature of the black hole living in an AdS space-time. This involves an intriguing and non-trivial twist of the “classic” consideration explaining Hawking radiation. In Hawking's computation one is dealing with quantised fields living in the classical black-hole space-time, whereas in the AdS bulk everything is strictly classical and zero temperature. Instead, via a remarkably elegant construction it is easy to understand that the black-hole bulk geometry “projects” onto the boundary system a finite temperature that is coincident with the Hawking temperature one would find in a bulk with quantised fields.

Having identified the dictionary rule that finite temperature is encoded by the bulk black-hole geometry, it turns out that these black holes also encode in an impeccable way for all the thermodynamics principles governing thermal equilibrium physics. This direct map of the “rules of black holes” to the thermodynamics of a real physical system with microscopic degrees of freedom is why the AdS/CFT correspondence manifests the holographic principle explained in the preceding chapters. The most poignant aspect hereof, as we will also discuss in the first section, is the identification of the Bekenstein–Hawking black-hole entropy with the entropy of microscopic configurations of the boundary field theory.

However, holographic thermodynamics is a lot more powerful than these classic black-hole thermodynamics notions. In later chapters we will show that AdS black holes are very different from their more familiar featureless all-engorging flat-space cousins. AdS black holes are actually able to describe rather rich, reallife phase diagrams of the matter in the boundary. In section 6.2 we will highlight a historically important example of such a phase diagram: the “Hawking–Page” confinement–deconfinement phase transition in a finite volume as discovered early on by Witten [211].