A computer microprocessor has a housing portion with a recess formed in a top side wall thereof, and a die portion inset within the recess. Operating heat from the die is removed by heat dissipation apparatus which automatically adapts to variations in the microprocessor bond line thickness and includes a sheet metal EMI shield wall overlying the microprocessor and having a condensing end portion of a thermosyphoning heat pipe secured to its bottom side, with an evaporating end portion of the heat pipe overlying the die recess and being resiliently deflectable toward a phase change thermal pad mounted on the top side of the die and inset into the housing die recess. A laterally Enlarged heat sink section is integrally formed on the evaporating end portion and has a flat bottom side and an arcuate top side. The integral heat sink section is resiliently deflected into the die recess, and has its flat bottom side resiliently pressed against the thermal interface pad, by a spring plate portion of a clamping structure downwardly engaging the arcuate top side of the heat sink section. During operator of the microprocessor, die heat is transferred to the metal shield wall sequentially through the thermal pad and the heat pipe, and then dissipated from the shield wall using fan-generated cooling air flowed along the shield wall.