A fundamental result about the dynamics and geometry of hyperbolic manifolds is Besson-Courtois-Gallot's entropy inequality. The volume entropy of a Riemannian metric measures the growth rate of geodesic balls in the universal cover. The result says that given a closed hyperbolic manifold (M,g_0), the metric with same volume as g_0 which minimizes the volume entropy is the hyperbolic one. We will discuss the corresponding stability problem: if a volume normalized metric g has entropy close to that of the hyperbolic metric g_0, is it true that g is "close" to g_0? We will give a positive answer, whose proof involves an area-minimization problem of independent interest. We will also sketch some intriguing examples suggesting that our answer may be optimal.