We prove the topological expansion for the cubic log–gas partition function $$\begin{aligned} Z_N(t)= \int _\Gamma \cdots \int _\Gamma \prod _{1\le j<k\le N}(z_j-z_k)^2 \prod _{k=1}^Ne^{-N\left( -\frac{z^3}{3}+tz\right) }\mathrm{dz}_1\cdots \mathrm{dz}_N, \end{aligned}$$ Z N ( t ) = ∫ Γ ⋯ ∫ Γ ∏ 1 ≤ j < k ≤ N ( z j - z k ) 2 ∏ k = 1 N e - N - z 3 3 + t z dz 1 ⋯ dz N , where t is a complex parameter and $$\Gamma $$ Γ is an unbounded contour on the complex plane extending from $$e^{\pi \mathrm{i}}\infty $$ e π i ∞ to $$e^{\pi \mathrm{i}/3}\infty $$ e π i / 3 ∞ . The complex cubic log–gas model exhibits two phase regions on the complex t-plane, with one cut and two cuts, separated by analytic critical arcs of the two types of phase transition: split of a cut and birth of a cut. The common point of the critical arcs is a tricritical point of the Painlevé I type. In the present paper we prove the topological expansion for $$\log Z_N(t)$$ log Z N ( t ) in the one-cut phase region. The proof is based on the Riemann–Hilbert approach to semiclassical asymptotic expansions for the associated orthogonal polynomials and the theory of S-curves and quadratic differentials.