Write LaTeX3 variable to aux file and recover it

Let's assume you have two integer variables \g_maxmaier_foo_int and \l_maxmaier_bar_int and that, at end document, you want to save the setting

\int_gset:Nn \g_maxmaier_foo_int { \int_eval:n { \l_maxmaier_bar_int } }

in the aux file, so that the value of the foo variable is set at the beginning of the next run.

You have to surround this in \ExplSyntaxOn and \ExplSyntaxOff, otherwise the commands would not be interpreted correctly. Note also that the setting to the foo variable must be global, because the aux file is read inside a group.

\documentclass{article}

\usepackage{xparse}

\ExplSyntaxOn
\int_new:N \g_maxmaier_foo_int
\int_new:N \l_maxmaier_bar_int

\NewDocumentCommand{\setbar}{m}
 {
  \int_set:Nn \l_maxmaier_bar_int { #1 }
 }
\NewDocumentCommand{\showfoo}{}
 {
  \int_to_arabic:n { \g_maxmaier_foo_int }
 }

\AtEndDocument
 {
  \iow_now:cx { @auxout }
   {
    \token_to_str:N \ExplSyntaxOn
    ^^J
    \int_gset:Nn \g_maxmaier_foo_int { \int_eval:n { \l_maxmaier_bar_int } }
    ^^J
    \token_to_str:N \ExplSyntaxOff
   }
 }
\ExplSyntaxOff

\begin{document}

The value of \texttt{foo} is \showfoo

\setbar{42}

\end{document}

Contents of the aux file

\relax 
\ExplSyntaxOn
\int_gset:Nn \g_maxmaier_foo_int {42}
\ExplSyntaxOff

---

Some comments.

1. Adding \int_eval:n (or its synonym \int_use:n) is essential, otherwise LaTeX would write

   \int_gset:Nn \g_maxmaier_foo_int { \l_maxmaier_bar_int }
   

   which would be completely useless, because it would assign the value of \l_maxmaier_bar_int current at execution time, that is, when the .aux file is being read; we want instead to fix the value of the variable at writing time (in the previous LaTeX run).

2. For the same reason above, it's important to use \iow_now:cx (a variant for \iow_now:Nx) so its argument is fully expanded at writing time.

3. The token ^^J makes a new line when a list of tokens is written in the .aux file. It's the analog of \n in other programming languages.