  Experimental exploration of high-cost systems with safety constraints, common
in engineering applications, is a challenging endeavor. Data-driven models
offer a promising solution, but acquiring the requisite data remains expensive
and is potentially unsafe. Safe active learning techniques prove essential,
enabling the learning of high-quality models with minimal expensive data points
and high safety. This paper introduces a safe active learning framework
tailored for time-varying systems, addressing drift, seasonal changes, and
complexities due to dynamic behavior. The proposed Time-aware Integrated Mean
Squared Prediction Error (T-IMSPE) method minimizes posterior variance over
current and future states, optimizing information gathering also in the time
domain. Empirical results highlight T-IMSPE's advantages in model quality
through toy and real-world examples. State of the art Gaussian processes are
compatible with T-IMSPE. Our theoretical contributions include a clear
delineation which Gaussian process kernels, domains, and weighting measures are
suitable for T-IMSPE and even beyond for its non-time aware predecessor IMSPE.
