How does the precession of the Earth's apses affect climate? The gravitational influence of the gas giants (plus general-relativistic effects) are known to cause the apses of the Earth's orbit to precess. For an illustration of that process, see this image. Now, it's clear that apsidal precession combined with changes in orbital eccentricity will affect climate - at minimum, this changes the mean (year-averaged) Earth-Sun distance, which one would of course expect to have an effect on climate. 

I'm not really following how apsidal precession alone is supposed to affect climate, though. Let us suppose that the Earth's "rotational parameters" (e.g. axial tilt) are held constant, and that the orbit of the Earth is static except for apsidal precession (i.e. no changes in eccentricity, no orbital decay, etc.). 

In this case, the total insolation experienced by the Earth over the course of a year should be independent of how much the apses have precessed - indeed, an observer looking at the Sun-Earth system from above the ecliptic plane should not be able to distinguish between, say, a 5-degree precession of the apses and a 5-degree rotation of the observer in the opposite direction. Since the latter cannot affect the Earth's climate, we should neither expect the former to.

Clearly, I'm making some sort of mistake here, but I'm not sure where exactly my thinking has gone wrong. Where is my error?
During the 20th century, the global mean temperature in January was 12°C while it was 15.8°C in July. This is a bit paradoxical as insolation is currently about 7% higher during January than it is in July. So why is July warmer than January?

The solution to this apparent paradox and the answer your question (how does apsidal precession alone affect the climate) are one and the same: The uneven distribution of land and water between the northern and southern hemispheres. The northern hemisphere has about twice as much land as does the southern hemisphere. Continental climates exhibit much more seasonal variability than do maritime climates.

That worldwide insolation is greatest in January and least in July is a bit irrelevant. What is more relevant is the length of the seasons in the northern hemisphere. Right now, summer is longer than is winter in the northern hemisphere because perihelion is in early January. This situation will be reversed in ten thousand years when perihelion and northern hemisphere summer solstice coincide. Looking at precession only, glaciation is much more likely during periods where northern hemisphere summers are short and winters are long.