I think too it's possibly broken.
A good Display shows a line of black Blocks in the first line, and nothing in the second line when only power is connected.
There are 2 controllers on this Display, so there must be shown 2 black lines ( 1 + 3 ) and 2 blank lines ( 2 + 4 ).
The display will geht hot, when the initialising of the display controller was wrong, in false order, or the timing at the pins isn't OK while initialising or running ( Look for busy flag ) .
Such wrong using can damage the display, because the display controller will get hot.
I've sarched a litte bit.
The Display Manufactor ist Nan Ya Plastics Corporation. The Display Type is possible LMMK5x028x2S.
On your display is a label with LMKK5R028A2U.