Failure prediction models can be significantly beneficial in managing large-scale complex software systems, but their trustworthiness is severely affected by changes in the data over time, also known as concept drift. Thus, monitoring these models against concept drift and retraining them when data has changed becomes a crucial step in designing reliable failure prediction models. In this work, we assess the effects of monitoring failure prediction models over time using label-independent (unsupervised) drift detectors. We show that retraining based on unsupervised drift detectors instead of periodically reduces the cost of acquiring true labels without compromising accuracy. Furthermore, we propose a novel feature reduction for unsupervised drift detectors and an evaluation pipeline that practitioners can employ to select the most suitable unsupervised drift detector for their application.