Relative entropy, as a divergence metric between two distributions, can be used for offline change-point detection and extends classical methods that mainly rely on moment-based discrepancies. To build a statistical test suitable for this context, we study the distribution of empirical relative entropy and derive several types of approximations: concentration inequalities for finite samples, asymptotic distributions, and Berry--Esseen bounds in a pre-asymptotic regime. For the latter, we introduce a new approach to obtain Berry--Esseen inequalities for nonlinear functions of sum statistics under some convexity assumptions. Our theoretical contributions cover both one- and two-sample empirical relative entropies. We then detail a change-point detection procedure built on relative entropy and compare it, through extensive simulations, with classical methods based on moments or on information criteria. In particular, we show that our method has a high power for many types of change-points. Finally, we illustrate the practical relevance of our method on a real dataset involving time series of volatility of stock indices, namely several realized volatility series as well as the VIX. In this application, we define the change-point as a modification of the multivariate distribution of successive increments of volatility. Therefore, every detection of a change-point highlights a modification of the serial dependence of volatility and thus a modification of the way people should predict this risk measure.