It seems as though it is either going into some sort of protect mode or perhaps there is a problem with the power supply in the box.
I cannot find a service manual or schematic for the box online so you may just have to open the box and initially check the power board for any obvious component damage such as heat stressed components, bulging or blown capacitors and dry solder joints on the back of the board.
Here’s the ifixit Apex Digital DTV Converter Box Power supply Board replacement guide that may or may not be of some help. I don’t think that it is the same model as yours (possibly a DT-250A?) but hopefully it is similar enough to be of some help.
If you see something wrong and/or you’re not quite sure, post some close up images back here. Here’s how to do this 在已经存在的问题里加入图片