I dont know this chip, but I do know systems thermal design. The spec reads:
"The A3977 is supplied in a choice of two power packages, a 44-pin plastic PLCC with 3 internally-fused pins on each of four sides (suffix ED), and a thin (<1.2 mm), 28-pin TSSOP with an exposed thermal pad (suffix LP). Both packages are lead (Pb) free, with 100% matte tin leadframe plating."

Both packages are intended to be cooled by contact with the PCB, either through the mounting tabs (ED) or via the underside pad (LP)

For the ED package the relevant thermal info is here and for the LP device here. It is essential to cool them via sufficient PCB ground area for which the thermal resistance (junction to PCB) is 6degC/W (ED) or 2degC/W (LP). If you want to stick a heatsink on it then stick it to the ground areas.

So irrespective of whether you want/like heat going into the PCB or not CheekyMonkies is correct (other than for the ED pack its all 4 tabs that conduct heat and all 4 must be used), that is the way the chip is intended to be used.

For such a device I would expect the case to get too hot to touch as what you are touching is the back of the heatpipe (the chip is actually under the metal support). Adding a stick-on heatsink here will make you feel better but will have minimal imact on internal junction temperatures as the thermal resistance from junction to case is very high.

Anyway, the chip has thermal shutdown.. if it were unhappy it would shut down and since it seems thats not the case I wouldnt worry...

BTW if that picture is of a board using the chip in question then my gut feel is that it would appear not to have enough copper heatsink area but its hard to tell without measuring it...