The Watchtower of July 15, 1907 says in an article on the Tabernacle (p.4028) :
Without claiming that phrenology has reached a perfection of development - without claiming that any has learned to read accurately from the shape of the human skull the various traits of character therein represented, even while admitting that such a reading of character might be defective, and particularly so with those whose characters have been transformed by the renewing of their mind through the begettal of the holy Spirit - nevertheless we may admit that phrenology so far as understood fully corroborates the picture given us in the arrangement of the Tabernacle of Israel surrounded by the camp.
It then goes on to say that if we imagine the human skull spread out flat that the central part of it would correspond to the Tabernacle and its court. It's very strange but it is hardly an endorsement of phrenology.
There is also a reference to the shape of the brain in the March 15, 1913 Watchtower (p.5201) but that is not talking about phrenology.
The drawing power which the Almighty exercises over humanity is in different degrees. Some have a strong desire to worship God, others have a weak desire, and others have no desire at all. This difference is due to the shape of the brain. Mankind are born with differences in this respect.--Psa. 51:5
In the Watchtower of July 15, 1978 in an article on the Human Brain (pp.14-18) it says :
By the use of electrodes, areas of the cerebral cortex have been mapped out, showing what functions are performed and where. Some false beliefs have been removed, such as phrenology—the study of “character traits” by feeling bumps on the head. The shape of the skull is not determined by the shape of the cerebrum, nor is it possible to assign “character traits” to specific areas of the brain.
So, really not very much at all. I wouldn't be surprised if there was some mention of it in the Golden Age but couldn't find any.