We prove a quantitative central limit theorem for linear statistics of particles in the complex plane, with Coulomb interaction at any temperature. This generalizes works by Rider, Virag, Ameur, Hendenmalm and Makarov obtained for the integrable model at inverse temperature beta=2. Important tools are a multi scale analysis and the loop equation. This is joint work with Roland Bauerschmidt, Miika Nikula and Horng-Tzer Yau.