I'm Following below method to display menu and it is working in my local but when i put it live the template path is not getting it's is showing like
frontend/default/theme_name/template/ the page path is not getting.
You should override the _getHtml() method in Mage_Page_Block_Html_Topmenu class.
Here are the steps:
class Mage_Page_Block_Html_Topmenu_Custom extends Mage_Page_Block_Html_Topmenu
_getHtml() method from app/code/code/Mage/Page/Block/Html/Topmenu.php and customize it as needed
app/design/frontend/your_theme/default/layout/page.xml and replace
<block type="page/html_topmenu" name="catalog.topnav" template="page/html/topmenu.phtml"/> with <block type="page/html_topmenu_custom" name="catalog.topnav" template="page/html/topmenu.phtml"/>