Entropy-Area Law and Temperature of de Sitter Horizons from Modular Theory

We derive an entropy-area law for the future horizon of an observer in diamonds inside a static patch of de Sitter space-time, taking into account the back reaction of quantum matter fields. We prove the positivity and convexity of the relative entropy for coherent states using Tomita-Takesaki modular theory, from which the quantum null energy condition for diamonds follows. Furthermore, we show that the generalized entropy conjecture holds. Finally, we reveal that the local temperature that is measured by an observer at rest exhibits subleading quantum corrections with respect to the well known cosmological horizon temperature H/(2π).