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;
}