Climate variability occurs over wide ranges of spatial and temporal scales. It exhibits a complex spatial covariance structure, which depends on geographic location (e.g., tropics vs extratropics) and also consists of a superposition of (i) components with gradually decaying positive correlation functions and (ii) teleconnections that often involve anticorrelations. In addition, there are indications that the spatial covariance structure depends on frequency. Thus, a comprehensive assessment of the spatiotemporal covariance structure of climate variability would require an extensive set of statistical diagnostics. Therefore, it is often desirable to characterize the covariance structure by a simple summarizing metric that is easy to compute from datasets. Such summarizing metrics are useful, for example, in the context of comparisons between climate models or between models and observations. Here we introduce a frequency-dependent version of a simple measure of the effective spatial degrees of freedom. The measure is based on the temporal variance of the global average of some climate variable, and its novel aspect consists in its frequency dependence. We also provide a clear geometric interpretation of the measure. Its easy applicability is demonstrated using near-surface temperature and precipitation fields obtained from a paleoclimate model simulation. This application reveals a distinct scaling behavior of the spatial degrees of freedom as a function of frequency, ranging from monthly to millennial scales.