  We study hole probabilities of two-dimensional Coulomb gases with a general
potential and arbitrary temperature. The hole region $U$ is assumed to satisfy
$\partial U\subset S$, where $S$ is the support of the equilibrium measure
$\mu$. Let $n$ be the number of points. As $n \to \infty$, we prove that the
probability that no points lie in $U$ behaves like $\exp(-Cn^{2}+o(n^{2}))$. We
determine $C$ in terms of $\mu$ and the balayage measure $\nu =
\mathrm{Bal}(\mu|_{U},\partial U)$. If $U$ is unbounded, then $C$ also involves
the Green function of $\Omega$ with pole at $\infty$, where $\Omega$ is the
unbounded component of $U$. We also provide several examples where $\nu$ and
$C$ admit explicit expressions: we consider several point processes, such as
the elliptic Ginibre, Mittag-Leffler, and spherical point processes, and
various hole regions, such as circular sectors, ellipses, rectangles, and the
complement of an ellipse. This work generalizes previous results of Adhikari
and Reddy in several directions.
