Wouldn't FDG-PET show hypometabolism of eccrine glands rather than hypermetabolism?

You're right to push: if the glands themselves are shut down, FDG-PET should show less uptake, not more — but the honest answer is that PET probably cannot see eccrine glands at all.

The mechanism cuts your way. FDG uptake tracks glucose use, and a gland that is not secreting is not burning much glucose. In the one anhidrosis case in these records, the sweat glands themselves were histologically normal and the failure was in the nerves supplying them [1], so the glands would be idle, not inflamed — idle tissue reads as hypometabolism or, more likely, as nothing distinguishable from background skin. Hypermetabolism would only be expected if the glands were actively inflamed or proliferating, which is a different disease. The functional-biology record supports the idea that sweating can be suppressed by local signaling (histamine blocking the acetylcholine-driven sweat response) without any structural gland damage [2], and suppressed-but-alive glands are not a hypermetabolic target.

Two caveats keep this from being a clean "yes, hypo." First, eccrine glands are millimeter-scale and scattered; PET resolution is far coarser, so the realistic finding is either normal-appearing skin or, at most, subtle diffuse regional differences — not a gland-level signal. Second, this retrieval contains no study of FDG-PET in anhidrosis or of eccrine-gland glucose uptake at all; the nearest records are the denervation case [1] and sweat-gland physiology [2]. So the hypometabolism prediction is first-principles reasoning, not a documented imaging finding.

My call: yes, if anything is visible, expect hypo- or isometabolism of the glands, not hyper — but the more likely result is a scan that cannot resolve them, which is why the gland-level question is better asked with a sweat test or biopsy than a PET. Confidence: moderate on the direction, low that PET could show it.