Cox regression with Lasso penalization is widely used for variable selection in time-to-event data, but reliable coefficient inference after selection remains difficult. We investigate bootstrap inference for the debiased Cox estimator after Cox Lasso selection. Two score-based procedures are considered: a wild bootstrap using independent multipliers and an Efron bootstrap using centered multinomial resampling weights. Both procedures keep the original Cox Lasso fit and debiasing matrix fixed and apply bootstrap weights to the score contributions that determine the first-order distribution of the debiased estimator. We establish asymptotic validity for both bootstrap schemes and study their finite-sample behavior in extensive simulations. The bootstrap intervals improve coverage accuracy over first-order debiased Wald intervals in many small- and moderate-sample settings, with the size of the improvement depending on the simulation setting and tuning rule. A SUPPORT2 application illustrates the procedures in practice.

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