Adaptive randomized clinical trials are of major interest when dealing with a time‐to‐event outcome in a prolonged observation window. No consensus exists either to define stopping boundaries or to combine values or test statistics in the terminal analysis in the case of a frequentist design and sample size adaptation. In a one‐sided setting, we compared three frequentist approaches using stopping boundaries relying on ‐spending functions and a Bayesian monitoring setting with boundaries based on the posterior distribution of the log‐hazard ratio. All designs comprised a single interim analysis with an efficacy stopping rule and the possibility of sample size adaptation at this interim step. Three frequentist approaches were defined based on the terminal analysis: combination of stagewise statistics (Wassmer) or of values (Desseaux), or on patientwise splitting (Jörgens), and we compared the results with those of the Bayesian monitoring approach (Freedman). These different approaches were evaluated in a simulation study and then illustrated on a real dataset from a randomized clinical trial conducted in elderly patients with chronic lymphocytic leukemia. All approaches controlled for the type I error rate, except for the Bayesian monitoring approach, and yielded satisfactory power. It appears that the frequentist approaches are the best in underpowered trials. The power of all the approaches was affected by the violation of the proportional hazards (PH) assumption. For adaptive designs with a survival endpoint and a one‐sided alternative hypothesis, the Wassmer and Jörgens approaches after sample size adaptation should be preferred, unless violation of PH is suspected.