The following has worked well for me:

use IO::String;
...
my $add_io = IO::String->new;
my $old_handle = select($add_io);
ok(
  $app->run,
  'ran app'
);
select($old_handle);

like(
  ${$add_io->string_ref},
  qr|Location:|,
  q|got application redirect|
);
undef $add_io;
[download]

This stores whatever $app->run sends to STDOUT in $add_io.