How make a table border thicker

One option is to insert the table into a tikz node with thick borders.

\documentclass[]{article}
\usepackage[table]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    \renewcommand\arraystretch{2.2}%
    \node[draw,ultra thick,inner sep=.4pt]{\begin{tabular}[]{|p{25mm}|c|c|c|}
      \hline
       \multicolumn{4}{|c|}{aaaaa} \\ % <===============================
      \hline
      \rowcolor{cyan}
      1 & 1e-1 & 1e-2 & 1e-2 \\
      \hline
      2 & 1e-1 & 1e-2 & 1e-2 \\
      \hline
      3 & 1e-1 & 1e-2 & 1e-2 \\
      \hline
      4 & 1e-1 & 1e-2 & 1e-2 \\
      \hline
      5 & 1e-1 & 1e-2 & 1e-2 \\
      \hline
    \end{tabular}};
\end{tikzpicture}

\end{document}

If I've well understood what you want, it's easy with boldline:

\documentclass[]{article}
\usepackage[table]{xcolor}
\usepackage{boldline} 

\begin{document}

\begingroup
\def\arraystretch{2.2}%
\begin{tabular}[]{V{4}p{25mm}|c|c|cV{4}}
  \hlineB{4}
   \multicolumn{4}{V{4}cV{4}}{aaaaa} \\ % <===============================
  \hline
  \rowcolor{cyan}
  1 & 1e-1 & 1e-2 & 1e-2 \\
  \hline
  2 & 1e-1 & 1e-2 & 1e-2 \\
  \hline
  3 & 1e-1 & 1e-2 & 1e-2 \\
  \hline
  4 & 1e-1 & 1e-2 & 1e-2 \\
  \hline
  5 & 1e-1 & 1e-2 & 1e-2 \\
  \hlineB{3}
\end{tabular}
\endgroup

\end{document} 

You can place the entire tabular inside an \fbox:

\documentclass{article}

\usepackage[table]{xcolor}
\usepackage{booktabs}

\begin{document}

\begingroup
\renewcommand\arraystretch{2.2}%
\setlength{\fboxsep}{0pt}%
\setlength{\fboxrule}{2.6pt}%
\fbox{%
  \begin{tabular}{|p{25mm}|c|c|c|}
    \hline
    \multicolumn{4}{|c|}{aaaaa} \\
    \hline
    \rowcolor{cyan}
    1 & 1e-1 & 1e-2 & 1e-2 \\
    \hline
    2 & 1e-1 & 1e-2 & 1e-2 \\
    \hline
    3 & 1e-1 & 1e-2 & 1e-2 \\
    \hline
    4 & 1e-1 & 1e-2 & 1e-2 \\
    \hline
    5 & 1e-1 & 1e-2 & 1e-2 \\
    \hline
  \end{tabular}%
}
\endgroup

\bigskip

\begin{tabular}{ p{25mm} *{3}{c} }
  \toprule
  \multicolumn{4}{c}{aaaaa} \\
  \midrule
  \rowcolor{cyan}%
  1 & 1e-1 & 1e-2 & 1e-2 \\
  2 & 1e-1 & 1e-2 & 1e-2 \\
  3 & 1e-1 & 1e-2 & 1e-2 \\
  4 & 1e-1 & 1e-2 & 1e-2 \\
  5 & 1e-1 & 1e-2 & 1e-2 \\
  \bottomrule
\end{tabular}

\end{document}

I'd be more inclined to use booktabs to set the table: