Hypothyroidism is known to cause a sluggish metabolism, which frequently results in elevated lipid levels. Over time, lipid excess along with the weight gain that can occur in hypothyroidism, can lead to a disease known as fatty liver.
Liver tissue that's clogged with fat deposits eventually fails to function and it can become cirrhotic. This is the first step leading to liver cancer.
In an article on ASHTA on this topic Valerie describes this study and explains how hypothyroidism causes an increased risk of liver cancer.