Heat Pumps and Electric Heating - Heat pumps and closed doors

I have a Trane heat pump. The inlet vent is located at the end of a hallway. 4 rooms in that hallway. All of the doors except one remain closed in that hallway. The third room has the vent closed. One of the rooms (one closest to the inlet vet) Just recently started remaining closed. I was told that having the doors closed is bad for a heat pump and I should leave them open. I have noticed an increase in my power bill the last couple months. My question is could this increase be due to having the doors closed. Also we don't want to open the doors because we have cats and don't want them in the rooms. Can I vent the doors at the bottom in some way to accomplish the proper air exchange that is needed?

Thanks for your help!

Normally in your set-up the bottoms of the doors should be cut off 1 1/2" - 2" for proper return flow.

Not allowing proper air flow through the dwelling could make your heat pump run longer causing higher bills. Yes, you can cut out a section of each door and add return louvers to the bottoms of the door. It will of course reduce the door's sound barrier.

If you have pets make sure you are vigilant about cleaning your filter. That too will make the system work harder.

I faced the same issue when replacing my 20 year old Lennox units. The second floor return was a single return in the hallway that served all four bedrooms. I solved the issue by adding two return ducts to the room closest to the air handler and adding a feed-through return air vent to the MBR. The 4th BR has a single inlet, and I cut the door to allow about 1" below it to give it sufficient airflow.

This really made a difference in how hard the variable speed air handler was working, it slowed things down a bunch.