I have a house that is two years old. In the master bedroom the closet has a vent that blows warm air, but the closet is freezing cold. If I keep the door of the closet open. The closet temperature is the same as the room. What do I need to do to keep the closet warm without the door open?

The only quick fix would cut a hole in the drywall above the door on both sides of the wall. Cut some 2x4s just to frame in the opening ans slip them inside the cut out frame. Use a drywall saw and be very cautious of any wires in the wall. I would cut the inside wall first that way if you see a wire and Need to adjust the square hole and move that hole a bit then a slight repair in the closet is easy and not seen
Get some square louvered grills from the big box stores to try and match your heat grills. Use this for a template to make the cuts. When you are done it will look like part of the heating system and not stick out. If the closet has an exposed sidewall and nothing is in that wall put grill there first. And if you want more heat flow into the closet add another one above door or to the side.
The ceiling heat has a chance to get into the closet.

You can replace the door with a louvered bi fold or a singe louvered door.
If the closet is getting real cold its possible there is a problem with the insulation. But most closet get colder on a outside wall and no airflow of heat gets in.
Grill is the quick easy fix.

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Signed 21 Boat

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Insulate the ceiling. It sounds like theirs nothing above ,but attic.

I do agree that you must have inadequate insulation. You have a hot air register but it still is freezing,bad exterior wall/ceiling insulation is the problem. Also, if warm air blows in but has nowhere to go it won't let much warm air in. You need to make a provision to allow the cold air to sink to the floor and be expelled into the room to make way for the volume of hot air to be unrestricted.