Housing Efficiency Standards Act of 2020 This bill updates energy efficiency requirements for new construction or rehabilitation of certain federally assisted housing, as well as for new construction of manufactured housing. Specifically, new construction or rehabilitation of certain federally assisted housing must comply with recently updated energy efficiency standards issued by certain industry bodies. Current law requires compliance with older versions of these standards. The Department of Housing and Urban Development and the Department of Agriculture may establish additional or different energy efficiency requirements if they are at least as stringent as the otherwise applicable standards. Additionally, new construction of manufactured housing must comply with standards issued by certain industry bodies or by the Department of Energy.