It is a heavily customized style so I'm not 100% sure that is "FORUMHOME" AKA forum.php but assuming it is you will want to edit the template
forumhome_forumbit_level2_post
If you look at that template you see that the status icon image (the image that changes depending on new posts or not) is built using the following code:
Code:
{vb:raw forum.imageprefix}forum_{vb:raw forum.statusicon}-48.png
Basically this means that {vb:raw forum.statusicon} is equal to "old" when there are no new posts and "new" when there are new posts (if you look at their file names you see they are forum_old-48.png and forum_new-48.png respecfully)
This means we can check the value of the variable in a template conditional and alter the code based on what the value is.
In a default template the code looks like this- it may be different on your custom style, but for me it is the 2nd line of the template
Code:
<div class="forumrow table">
Now edit it to look like this:
Code:
<div class="forumrow table" <vb:if condition="$forum['statusicon'] == 'new'">style="background-color:yellow;"</vb:if>>
That will make the background of any of the forums with new posts yellow (change the color as you wish as well)
Tested this against a default style.