We consider the estimation of sensitivity indices based on divergence measures such as Hellinger distance. For sensitivity analysis of complex models, these divergence-based indices can be estimated by Monte-Carlo sampling (MCS) in combination with kernel density estimation (KDE). In a direct approach, the complex model must be evaluated at every input point generated by MCS, resulting in samples in the input-output space that can be used for density estimation. However, if the computational cost of the complex model strongly limits the number of model evaluations, this direct method gives large errors. We propose to use Gaussian process (GP) surrogates to increase the number of samples in the combined input-output space. By enlarging this sample set, the KDE becomes more accurate, leading to improved estimates. To compare the GP surrogates, we use a surrogate constructed by samples obtained with stochastic collocation, combined with Lagrange interpolation. Furthermore, we propose a new estimation method for these sensitivity indices based on minimum spanning trees. Finally, we also propose a new type of sensitivity indices based on divergence measures, namely direct sensitivity indices. These are useful when the input data is dependent.