We study large deviations for a Markov process on curves in $\mathbb{Z}^2$ mimicking the motion of an interface. Our dynamics can be tuned with a parameter $β$, which plays the role of an inverse temperature, and coincides at $β$ = $\infty$ with the zero-temperature Ising model with Glauber dynamics, where curves correspond to the boundaries of droplets of one phase immersed in a sea of the other one. We prove that contours typically follow a motion by curvature with an influence of the parameter $β$, and establish large deviations bounds at all large enough $β$ < $\infty$. The diffusion coefficient and mobility of the model are identified and correspond to those predicted in the literature.