  For the two-dimensional one-component Coulomb plasma, we derive an asymptotic
expansion of the free energy up to order $N$, the number of particles of the
gas, with an effective error bound $N^{1-\kappa}$ for some constant $\kappa >
0$. This expansion is based on approximating the Coulomb gas by a quasi-free
Yukawa gas. Further, we prove that the fluctuations of the linear statistics
are given by a Gaussian free field at any positive temperature. Our proof of
this central limit theorem uses a loop equation for the Coulomb gas, the free
energy asymptotics, and rigidity bounds on the local density fluctuations of
the Coulomb gas, which we obtained in a previous paper.
