% arara: pdflatex
% arara: makeindex
% arara: pdflatex
\documentclass{article}
\usepackage{amsthm}
\usepackage{thmtools}
\usepackage[thickness=1.75pt, vskip=0em]{thmbox} % optional 1/2
\makeatletter
\newif\ifToIndex
\ToIndextrue % Default
\declaretheoremstyle[
notebraces={}{},
headpunct={\normalfont.\hspace{1.5em}},
%notefont=\normalfont,
headformat={\NAME\,%
\NUMBER\,%
\let\thmt@space\@empty{\normalfont(\NOTE)}%
\ifToIndex\index{\NOTE}\fi% <-------!!!!
},
preheadhook=\hspace{\baselineskip}\begin{leftbar}, postfoothook=\end{leftbar}, % optional 2/2