Database queries present a potential privacy risk to users, as they may disclose sensitive information about the person issuing the query. Consequently, privacy preserving query processing has gained significant attention in the literature, and numerous techniques have been proposed that seek to hide the content of the queries from the database server. Secure hardware-assisted private information retrieval (PIR) is currently the only practical solution that can be leveraged to build algorithms that provide perfect privacy. Nevertheless, existing approaches feature amortized page retrieval costs and, for large databases, some queries may lead to excessive delays, essentially taking the database server offline for large periods of time. In this paper, we address this drawback and introduce a novel approach that sacrifices some degree of privacy in order to provide fast and constant query response times. Our method leverages the internal cache of the secure hardware to constantly reshuffle the database pages in order to create sufficient uncertainty regarding the exact location of an arbitrary page. We give a formal definition of the privacy level of our algorithm and illustrate how to enforce it in practice. Based on the performance characteristics of the current state-of-the-art secure hardware platforms, we show that our method can provide low page access times, even for very large databases.