One of the most critical hypothesis in BEPU studies is the choice of the probability distributions of uncertain input variables which are propagated through a numerical model. Depending on the method used for this choice, the probability density functions (pdf) for these variables are established with various levels of confidence, not always well known. Hence, bringing stringent justifications to the BEPU approach requires quantifying the impact of this second-level uncertainty (i.e. the uncertainty on the input variable pdf). To solve this problem, this paper deepens the robustness analysis based on the "Perturbed-Law based sensitivity Indices" (PLI). The PLI quantifies the impact of a perturbation of an input pdf on the quantity of interest (as a quantile of a model output or a safety margin) in the BEPU study. The mathematical formalism of the PLI is firstly applied to two particular quantities of interest: the quantile and the superquantile (which is the mean of the pdf tail exceeding the quantile). For both quantities, the PLI can be easily computed using a unique Monte-Carlo sample (ensemble of code runs) containing model inputs and output. Then, numerical tests are developed in order to define validity criteria of a PLI-based robustness analysis. The method is finally illustrated to the study of the thermal-hydraulic responses of a test facility during a cold leg Intermediate Break Loss Of Coolant Accident (IBLOCA) modelled using the calculation code CATHARE.