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When the path to temp-directory you have entered in your Joomla Global Configuration does not exist or is not writeable Joomla uses your site's root as the fallback tmp directory and reports that to its extensions. Akeeba Backup is therefore told by Joomla that the temp-directory is your site root and refuses to proceed with a backup since this is a condition that leads to partial, practically useless backups.

Please consult with your host again and ask them to check very carefully if the path to the temp-directory defined in your Global Configuration (therefore: in your site's configuration.php file):

1. Exists
2. Is within the open_basedir path (if open_basedir restrictions are in effect)
3. Has the correct ownership and permissions to be directly writeable by the PHP process used to serve your site's index.php and administrator/index.php files.

They will see that either 1 or 2 are false. In this case they can give you the correct path to put in the Global Configuration.

PS: To make sure that what I just said is indeed the case I deliberately modified the temp-directory of one of my development sites, putting in a path to a directory that doesn't exist. Sure enough, I got the same error message as you did. Sure enough, resetting the temp-directory path to something that exists and is writeable makes the error message go away.