The thermodynamic Euler equation for high-energy states of large-$N$ gauge theories is derived from the dependence of the extensive quantities on the number of colors $N$. This Euler equation relates the energy of the state to the temperature, entropy, number of degrees of freedom and its chemical potential, but not to the volume or pressure. In the context of the gauge/gravity duality we show that the Euler equation is dual to the generalized Smarr formula for black holes in the presence of a negative cosmological constant. We also match the fundamental variational equation of thermodynamics to the first law of black hole mechanics, when extended to include variations of the cosmological constant and Newton’s constant.

We study several aspects of higher-order gravities constructed from general contractions of the Riemann tensor and the metric in arbitrary dimensions. First, we use the fast-linearization procedure presented in [P. Bueno and P. A. Cano, arXiv:1607.06463] to obtain the equations satisfied by the metric perturbation modes on a maximally symmetric background in the presence of matter and to classify $\mathcal{L}(\mathrm{Riemann})$ theories according to their spectrum. Then, we linearize all theories up to quartic order in curvature and use this result to construct quartic versions of Einsteinian cubic gravity. In addition, we show that the most general cubic gravity constructed in a dimension-independent way and which does not propagate the ghostlike spin-2 mode (but can propagate the scalar) is a linear combination of $f(\mathrm{Lovelock})$ invariants, plus the Einsteinian cubic gravity term, plus a new ghost-free gravity term. Next, we construct the generalized Newton potential and the post-Newtonian parameter $\gamma$ for general $\mathcal{L}(\mathrm{Riemann})$ gravities in arbitrary dimensions, unveiling some interesting differences with respect to the four-dimensional case. We also study the emission and propagation of gravitational radiation from sources for these theories in four dimensions, providing a generalized formula for the power emitted. Finally, we review Wald’s formalism for general $\mathcal{L}(\mathrm{Riemann})$ theories and construct new explicit expressions for the relevant quantities involved. Many examples illustrate our calculations.

We study the semi-classical thermodynamics of two-dimensional de Sitter space (dS₂) in Jackiw-Teitelboim (JT) gravity coupled to conformal matter. We extend the quasi-local formalism of Brown and York to dS₂, where a timelike boundary is introduced in the static patch to uniquely define conserved charges, including quasi-local energy. The boundary divides the static patch into two systems, a cosmological system and a black hole system, the former being unstable under thermal fluctuations while the latter is stable. A semi-classical quasi-local first law is derived, where the Gibbons–Hawking entropy is replaced by the generalized entropy. In the microcanonical ensemble the generalized entropy is stationary. Further, we show the on-shell Euclidean microcanonical action of a causal diamond in semi-classical JT gravity equals minus the generalized entropy of the diamond, hence extremization of the entropy follows from minimizing the action. Thus, we provide a first principles derivation of the island rule for U(1) symmetric dS₂ backgrounds, without invoking the replica trick. We discuss the implications of our findings for static patch de Sitter holography.

The static patch of de Sitter spacetime and the Rindler wedge of Minkowski spacetime are causal diamonds admitting a true Killing field, and they behave as thermodynamic equilibrium states under gravitational perturbations. We explore the extension of this gravitational thermodynamics to all causal diamonds in maximally symmetric spacetimes. Although such diamonds generally admit only a conformal Killing vector, that seems in all respects to be sufficient. We establish a Smarr formula for such diamonds and a ``first law" for variations to nearby solutions. The latter relates the variations of the bounding area, spatial volume of the maximal slice, cosmological constant, and matter Hamiltonian. The total Hamiltonian is the generator of evolution along the conformal Killing vector that preserves the diamond. To interpret the first law as a thermodynamic relation, it appears necessary to attribute a negative temperature to the diamond, as has been previously suggested for the special case of the static patch of de Sitter spacetime. With quantum corrections included, for small diamonds we recover the ``entanglement equilibrium'' result that the generalized entropy is stationary at the maximally symmetric vacuum at fixed volume, and we reformulate this as the stationarity of free conformal energy with the volume not fixed.

By respecting the conformal symmetry of the dual conformal field theory, and treating the conformal factor of the Anti-de Sitter boundary as a thermodynamic parameter, we formulate the holographic first law that is exactly dual to the first law of extended black hole thermodynamics with variable cosmological constant but fixed Newton’s constant.

We study the holographic dual of the extended thermodynamics of spherically symmetric, charged AdS black holes in the context of the AdS/CFT correspondence. The gravitational thermodynamics of AdS black holes can be extended by allowing for variations of the cosmological constant and Newton’s constant. In the dual CFT this corresponds to including the central charge C and its chemical potential μ as a new pair of conjugate thermodynamic variables, in addition to the standard pairs: temperature vs. entropy (T, S), electric potential vs. charge ( $$ \overset{\sim }{\Phi},\overset{\sim }{Q} $$Φ ~ , Q ~ ) and field theory pressure vs. volume (p, $$ \mathcal{V} $$V ). For the (grand) canonical ensembles at fixed ( $$ \overset{\sim }{Q},\mathcal{V},C $$Q ~ , V , C ), ( $$ \overset{\sim }{\Phi},\mathcal{V},C $$Φ ~ , V , C ) and ( $$ \overset{\sim }{Q},\mathcal{V},\mu $$Q ~ , V , μ ), based on the holographic dictionary, we argue the CFT description of charged AdS black holes contains either critical phenomena or interesting phase behaviour. In the fixed ( $$ \overset{\sim }{Q},\mathcal{V},\mu $$Q ~ , V , μ ) we find a new zeroth-order phase transition between a high- and low-entropy phase at some μ-dependent temperature. Finally, we point out there is no critical behaviour in the fixed p ensembles, i.e. there is no p − $$ \mathcal{V} $$V criticality, and hence the CFT state dual to a classical charged black hole cannot be a Van der Waals fluid. Whether or not this phase structure is supported by CFT computations remains an interesting open question.