Let (X,d) be a separable geodesic Gromov-hyperbolic space, o∈X a basepoint and μ a countably supported non-elementary probability measure on Isom(X). Denote by z n the random walk on X driven by the probability measure μ. Supposing that μ has a finite exponential moment, we give a second-order Taylor expansion of the large deviation rate function of the sequence 1 nd(z n ,o) and show that the corresponding coefficient is expressed by the variance in the central limit theorem satisfied by the sequence d(z n ,o). This provides a positive answer to a question raised in [6]. The proof relies on the study of the Laplace transform of d(z n ,o) at the origin using a martingale decomposition first introduced by Benoist–Quint together with an exponential submartingale transform and large deviation estimates for the quadratic variation process of certain martingales.