We obtain a general sublinear upper bound on the multiplicity of Laplacian eigenvalues for negatively curved surfaces. In particular, this yields progress on a longstanding conjecture by Colin de Verdière [Colin de Verdière, 1986]. Our proof relies on a trace argument for the heat kernel, and on the idea of leveraging an r-net in the surface to control this trace. This last idea was introduced in [Jiang-Tidor-Yao-Zhang-Zhao, 2021] for similar spectral purposes in the context of graphs of bounded degree. Our method is robust enough to also yield an upper bound on the "approximate multiplicity" of eigenvalues, i.e., the number of eigenvalues in windows of size 1 / log^{\kappa}(g), \kappa > 0.