Recent research on mutual exclusion for shared-memory systems has focused on "local spin" algorithms. Performance is measured using the "remote memory references" (RMRs) metric. As common in recent literature, we consider a standard asynchronous shared memory model with N processes, which allows atomic read, write and compare-and-swap (short: CAS) operations. In such a model, the asymptotically tight upper and lower bound on the number of RMRs per passage through the Critical Section is Theta(log N) for the optimal deterministic algorithms (see Yang and Anderson,1995, and Attiya, Hendler and Woelfel, 2008). Recently, several randomized algorithms have been devised that break the Omega(log N) barrier and need only o(log N) RMRs per passage in expectation (see Hendler and Woelfel, 2010, Hendler and Woelfel, 2011, and Bender and Gilbert, 2011). In this paper we present the first randomized "abortable" mutual exclusion algorithm that achieves a sub-logarithmic expected RMR complexity. More precisely, against a weak adversary (which can make scheduling decisions based on the entire past history, but not the latest coin-flips of each process) every process needs an expected number of O(log N/ log log N) RMRs to enter end exit the critical section. If a process receives an abort-signal, it can abort an attempt to enter the critical section within a finite number of its own steps and by incurring O(log N/ log log N) RMRs.