I would add the category slug as a class to each category link, and change your class name to be more specific so other CSS doesn’t conflict on accident:
<div class="category-link">
<a href="#" class="<?= $cats[0]->slug ?>"><?= $cats[0]->name ?></a>
</div>
Then call the class with CSS:
.category-link .food {
background-color: red;
}