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 the static patch of de Sitter spacetime, taking into account the backreaction of quantum matter fields. We prove positivity and convexity of the relative entropy for coherent states using Tomita--Takesaki modular theory, from which the QNEC for diamonds follows. Furthermore, we show that the generalized entropy conjecture holds. Finally, we reveal that the local temperature which is measured by an observer at rest exhibits subleading quantum corrections with respect to the well-known cosmological horizon temperature $H/(2\pi)$.