Mutual exclusion (ME) is one of the most commonly used techniques to handle conflicts in concurrent systems. Traditionally, mutual exclusion algorithms have been designed under the assumption that a process does not fail while acquiring/releasing a lock or while executing its critical section. However, failures do occur in real life, potentially leaving the lock in an inconsistent state. This gives rise to the problem of \emph{recoverable mutual exclusion (RME)} that involves designing a mutual exclusion algorithm that can tolerate failures, while maintaining safety and liveness properties. One of the important measures of performance of any ME algorithm, including an RME algorithm, is the number of \emph{remote memory references (RMRs)} made by a process (for acquiring and releasing a lock as well as recovering the lock structure after a failure). The best known RME algorithm solves the problem for $n$ processes in sub-logarithmic number of RMRs, given by $\mathcal{O}(\frac{\log n}{\log \log n})$, irrespective of the number of failures in the system. In this work, we present a new algorithm for solving the RME problem whose RMR complexity gradually \emph{adapts} to the number of failures that have occurred in the system "recently". In the absence of failures, our algorithm generates only $\mathcal{O}(1)$ RMRs. Furthermore, its RMR complexity is given by $\mathcal{O}(\min\{ \sqrt{F}, \frac{\log n}{\log \log n} \})$ where $F$ is the total number of failures in the "recent" past. In addition to read and write instructions, our algorithm uses compare-and-swap (\CAS{}) and fetch-and-store (\FAS{}) hardware instructions, both of which are commonly available in most modern processors.