Magento – Form Key Problems After Changing Domains 1.9.2.1

Presumably that's because the visitor's session expired. The form key is tied to the session. When they submit the form, Magento sees they have no valid session, and so creates a new one (along with a new form key). That new key doesn't match the one submitted, so they get an error--but submitting again immediately after that failure would indeed work.

Solution? Raise your session lifetime.

1. Make sure cookies are staying around (I would recommend expiration date of a day or more). See: System > Configuration > Web > Session Cookie Management > Cookie Lifetime
2. Make sure PHP isn't cleaning out sessions prematurely. That means increasing the value of PHP setting session.gc_maxlifetime (by default, set to 24 minutes). You may have to contact your host to arrange this.

You need to some change at your code;

Call a frontend design by Mage::app()->loadArea('frontend');

then Call formkey block from current exiting layout $layout = Mage::getSingleton('core/layout');

get block by code:

$layout = Mage::getSingleton('core/layout');

//load default xml layout handle and generate blocks
$layout->getUpdate()->load('default');  // add default handler
$layout->generateXml()->generateBlocks();

//get the loaded head and header blocks and output
$BlockHtml = $layout->getBlock('BlockName');

 echo $BlockHtml->toHtml() 

For this case you will get same session form_key value:

Modified code:

<?php
require '../app/Mage.php';
umask(0);
Mage::app()->loadArea('frontend');
define('ROOT', Mage::getBaseUrl(Mage_Core_Model_Store::URL_TYPE_WEB));

$layout = Mage::getSingleton('core/layout');

//load default xml layout handle and generate blocks
$layout->getUpdate()->load('default');
$layout->generateXml()->generateBlocks();

//get the loaded head and header blocks and output
$formkeyBlock = $layout->getBlock('formkey');
$key= Mage::getSingleton('core/session')->getFormKey(); 
?>
  <form action="mydomain/store/customer/account/loginPost/" method="post">
    <input name="form_key" value="<?php echo $key; ?>" type="hidden">
    <input type="text" name="login[username]" value="username" />
<?php echo $formkeyBlock->toHtml()  ?>
    <input type="password" name="login[password]" value="password" />

    <input type="submit" name="submit" value="submit" />

    </form>