Don't.
Instead, save the variable to a separate state file using Data::Dumper, then eval it back into your program. This is rather common, actually. I wrote a package once to create an OO persistent symbol table, I don't have it handy, but here is some basic (non-OO and un-tested) code to start from:

use strict;
my $persistentVar = $defaultvalue;
if( -e $persistentStateFile )
{
  open FH, $persistentStateFile 
    or die "Failed to open '$persistentStateFile': $!";
  local $/, $@;
  my $ps = <FH>;
  eval $ps;
  die $@ if $@;
}

END{
  open FH, "> $persistentStateFile"
    or warn "Failed to open '$persistentStateFile': $!" and exit(1);
  print FH Data::Dumper->Dump( [$refToPersistentVar], ['nameofpersiste
+ntvar'] ) 
    or warn "Failed to print persistent data, $!" and exit(1);
  close FH 
    or warn "Failed to close $persistentStateFile, $!" and exit(1);
}
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However, if you really wanted to replace the currently running script on exit, then do something like:

END {
    open FH, "< $0" or warn "Failed to preserve state... couldn't open
+ '$0', $!" and exit(1);
    local $/;
    $_ = <FH>;
    s/my \$var=$oldvalue/my \$var=$var/m;
    open FH, "> $0" or warn "Failed to preserve state... couldn't writ
+e to '$0', $!" and exit(1);
    print FH $_;
    close $file;
}
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I don't know if that would work, I wouldn't even try. But if you want, go for it.