Clinical trialists often face the challenge of balancing scientific questions with other design features, such as improving efficiency, minimizing exposure to inferior treatments, and simultaneously comparing multiple treatments. While Bayesian response adaptive randomization (RAR) is a popular and effective method for achieving these objectives, it is known to have large variability and a lack of explicit theoretical results, making its use in clinical trials a subject of concern. It is desirable to propose a design that targets the same allocation proportion as Bayesian RAR and achieves the above objectives but addresses the concerns over Bayesian RAR. We propose the frequentist doubly adaptive biased coin designs (DBCD) targeting ethical allocation proportions from the Bayesian framework to satisfy different objectives in clinical trials with time‐to‐event endpoints. We derive the theoretical properties of the proposed adaptive randomization design and show through comprehensive numerical simulations that it can achieve ethical objectives without sacrificing efficiency. Our combined theoretical and numerical results offer a strong foundation for the practical use of RAR in real clinical trials.