The Act, which was given presidential assent by Mr Museveni in February, calls for offenders to be sentenced to 14 years in prison and makes it a criminal offence to not report someone for being gay.
Last week, the U.S. announced that it had cut aid to Uganda, cancelled a planned military exercise with the country and banned Ugandan citizens involved in human rights abuses from entering the United States.
In her book, Clinton also writes about her anger at the murder of Ugandan gay rights campaigner David Kato in 2011.
“David was killed in what police said was a robbery but it was more likely an execution. I was appalled that the police and government had done little to protect David after public calls for his murder. But this was about more than police incompetence. It was the result of a nationwide campaign to suppress LGBT people by any means necessary, and the government was part of it."