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Paleoclimatology -- the study of past climate -- is relevant beyond climate science itself, such as in archaeology and anthropology for understanding past human dispersal. Information about the Earth's paleoclimate comes from simulations of physical and biogeochemical processes and from proxy records found in naturally occurring archives. Climate-field reconstructions (CFRs) combine these data into a statistical spatial or spatiotemporal model. To date, there exists no consensus spatiotemporal paleoclimate model that is continuous in space and time, produces predictions with uncertainty, and can include data from various sources. A Gaussian process (GP) model would have these desired properties; however, GPs scale unfavorably with data of the magnitude typical for building CFRs. We propose to build on recent advances in sparse spatiotemporal GPs that reduce the computational burden by combining variational methods based on inducing variables with the state-space formulation of GPs. We successfully employ such a doubly sparse GP to construct a probabilistic model of European paleoclimate from the Last Glacial Maximum (LGM) to the mid-Holocene (MH) that synthesizes paleoclimate simulations and fossilized pollen proxy data.