Center text between double horizontal rules

You can use the same idea, but leaders don't work here, so I prepare a vbox with the double rules after computing their width.

Somewhat arbitrarily, the space between the rules is four times their thickness.

\documentclass{article}

\newcommand{\textbetweendoublerules}[2][.4pt]{%
  \par\addvspace{\topsep}
  \noindent\makebox[\textwidth]{%
    \sbox0{\quad#2\quad}%
    \dimen0=.5\dimexpr\ht0+#1\relax
    \dimen2=-.5\dimexpr\ht0-#1\relax
    \dimen4=.5\dimexpr\textwidth-\wd0\relax
    \setbox2=\vbox to \ht0{%
      \vss
      \hrule width \dimen4 height #1
      \kern 4\dimexpr#1\relax
      \hrule width \dimen4 height #1
      \vss
    }%
    \copy2 \box0 \box2
  }\par\nopagebreak\addvspace{\topsep}%
}

\begin{document}

\textbetweendoublerules{abc}

\end{document}

I would go a different route and place the text in a coloured box over the lines. This way you don't have to worry about the centering.

(The code below is just a prove of concept, please adjust the lengths to your needs)

\documentclass{article}
\usepackage{lipsum}
\usepackage{xcolor}

\newcommand{\textbetweenrules}[1]{%
    \begin{center}
        \setlength{\fboxsep}{0.4cm}
        \rule{\textwidth}{1pt}\\[-0.3cm]
        \rule{\textwidth}{1pt}\\[-0.65cm]
        \colorbox{white}{#1}
    \end{center}
}

\begin{document}

\lipsum[1]

\textbetweenrules{Text}

\lipsum[2]

\textbetweenrules{longer text text text}

\lipsum[3]

\end{document}

It might be overkill to load TikZ for that, but if you load it anyway, you might consider this code:

\documentclass{article}
\usepackage{lipsum}
\usepackage{tikz}


\newcommand{\RuleText}[1]{%
\centerline{\tikz{\draw[double,line width=1pt,double distance between line
centers=3pt] (0,0)--(\linewidth,0) node[midway,fill=white]{#1};}}}

\begin{document}

\lipsum[1]

\RuleText{koala bear}

\lipsum[2]

\RuleText{marmot}

\lipsum[3]

\end{document}