It's called a rack.
A hob is a shelf at the side or back of a fireplace.
The shelf over a fireplace is called a mantle.
mantle. Mantel Mantelpiece Mantelshelf
The top part of a fireplace mantle is called the mantel shelf. It is the horizontal ledge that extends across the top of the fireplace to display decorative items.
A shelf on top of a fireplace is commonly referred to as a mantel. It is used for decorative purposes and can hold various items such as picture frames, candles, or seasonal decorations.
A hob.
chimney
A mantle.
hob
A shelf above a fireplace or something that covers or conceals
One can find shelves over the fireplace from Home Depot. One can go online to check with the shelf they want, then order it online or pick it up from store.
The corner of a fireplace is called an Inglenook!