Without an MWE in the question (document class, TOC related packages etc) it can be only a guess.
You could use package tocloft and add \cftpagenumbersoff{chapter} to the TOC.
Example:
\documentclass{report}
\usepackage{blindtext}% only for dummy text
\usepackage{tocloft}
\begin{document}
\tableofcontents
\blinddocument
\cleardoublepage
\setcounter{page}{1}
\addtocontents{toc}{\protect\cftpagenumbersoff{chapter}}% <- added
\addcontentsline{toc}{chapter}{Appendices}
\chapter*{Appendix 1: Room Temperature}
\addcontentsline{toc}{chapter}{Appendix 1: Room Temperature}
\blindtext
\end{document}

Or suggestion with package tocbasic:
\documentclass{report}
\usepackage{blindtext}% only for dummy text
\usepackage{tocbasic}
\newcommand\gobble[1]{}
\begin{document}
\tableofcontents
\blinddocument
\cleardoublepage
\setcounter{page}{1}
\addtocontents{toc}{\DeclareTOCStyleEntry[pagenumberformat=\protect\gobble]{tocline}{chapter}}
\addcontentsline{toc}{chapter}{Appendices}
\chapter*{Appendix 1: Room Temperature}
\addcontentsline{toc}{chapter}{Appendix 1: Room Temperature}
\blindtext
\end{document}

Or with \appendix:
\documentclass{report}
\usepackage{blindtext}% only for dummy text
\usepackage{tocbasic}
\newcommand\gobble[1]{}
\newcommand\appendixprefixintoc[1]{\appendixname~#1:\hfill}
\begin{document}
\tableofcontents
\blinddocument
\cleardoublepage
\appendix
\setcounter{page}{1}
\addtocontents{toc}{\DeclareTOCStyleEntry[pagenumberformat=\protect\gobble]{tocline}{chapter}}
\addcontentsline{toc}{chapter}{Appendices}
\addtocontents{toc}{\DeclareTOCStyleEntry[entrynumberformat=\protect\appendixprefixintoc,dynnumwidth]{tocline}{chapter}}
\chapter{Room Temperature}
\blindtext
\end{document}

