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Black holes are a cornerstone of popular science: the mystery surrounding them and the duality between their destructive behaviour and their almost eternal peacefulness has captured the eye of the public for years. We used to only think about black holes. A few years ago, the detection of gravitational waves allowed to hear them. Now, we can see them thanks to the Event Horizon Telescope. To the physicist, the black hole is a laboratory, a unique window into high-energy physics. The second half of the 20th century has seen many theoretical developments regarding black holes. A particularly active area was that of black hole thermodynamics, the idea that black holes are governed by laws analogous to those of thermodynamics. In that way, one can assign thermodynamical quantities to black holes, such as temperature or entropy - roughly speaking, the measure of disorder in a system. Thermodynamics contains the notion of phase transition: think of ice melting into water, or water becoming vapour. At the beginning of the eighties, Stephen Hawking and Don Page identified such a phenomenon between two different spacetimes: Anti de Sitter (AdS) and Schwarzschild-AdS. The latter contains a black hole, while the former does not. We can see this phase transition as radiation forming a black hole when a certain temperature is reached, like water would transform to ice at the freezing point. We know that Einstein's theory of General Relativity (GR) is not the final picture for gravity: at very high energies, the theory breaks down. This implies that a more general theory must exist, of which GR is a working instance at low energies. Supergravity combines GR with symmetry concepts from quantum mechanics, and is itself a low-energy realisation of string theory. As a generalisation of GR, the content of this theory is broader and includes higher-derivative corrections to GR. The goal of this thesis was to consider these higher-derivative corrections up to a reasonable order - in fact, the first order of correction to GR -, and see how they affect the Hawking-Page phase transition. Our particular model included two unknown parameters, so the corrections to the transition also depend on these parameters. Involving insight from quantum theories linked to AdS, the two parameters can be related, leaving us with one unknown. Once that corrections are included, there is no guarantee that the Hawking-Page phase transition still exists. The main result of this thesis is the determination of the range of the remaining parameter that allows a Hawking-Page transition. In particular, as we expect this coefficient to be very, very small due to his high-energy origin, we conclude that the Hawking-Page transition is inevitably contained in our higher-derivative model.