It suggests that in order for an organism to evolve a large brain without a significant increase in basal metabolic rate (as seen in humans), the organism must use less energy on other expensive tissues; the paper introducing the ETH suggests that in humans, this was achieved by eating an easy-to-digest diet and evolving a smaller, less energy-intensive gut.
They found that humans’ smaller relative gut size almost completely compensated for the metabolic cost of the larger brain.
[1] Anthropologists have been able to observe a dramatic contrast in relative brain size between humans and our great ape ancestors.
One such study supported the expensive tissue hypothesis and found a positive correlation between diet quality and brain size (as would be expected by the original paper), but it did note that there were exceptions among the species tested.
A 2018 study by Huang, Yu, and Liao investigated the possible effects of gut microbiota in the expensive tissue hypothesis among vertebrates.
These microbiotas have evolved to form mutually beneficial relationships with their hosts, and play important roles in immune function, nutrition, and physiology.
Additionally, the choice of the host is strongly associated with the diversification and complexity of the microbiota; for instance, the study illustrated that a diet high in fat increases the number of bacteria belonging to the phylum Bacteroidetes and decreases the number belonging to the Firmicutes in children's guts, and also theorized that diet quality is related to gut size.
Some evidence also suggests that vertebrates with large brains have evolved to balance out the energetic expenditure by trading off with gut size.
[6] However, after several investigations, the study did not find strong evidence to support the notion that brain size is negatively correlated with gut microbiota in vertebrates.
The result of the experiment concluded that the energy cost of encephalization might have played a role in the evolution of brain size in both endothermic as well as ectothermic vertebrates.