We report the measurement of the chemical potential of a two-dimension
al hole layer in wide temperature and density ranges. Due to their hea
vy mass, the holes are in a regime where the ratio of Coulomb to Fermi
energies r(s) is very high, thus facilitating the first detailed stud
y of the full temperature-dependent crossover from the strongly intera
cting degenerate regime to the classical ideal gas limit. The measured
low-temperature chemical potential and inverse compressibility are ne
gative and uninfluenced by temperature. At T approximate to 17 K they
simultaneously and abruptly become temperature-dependent for all carri
er concentrations. As the temperature is further raised, the inverse c
ompressibility becomes less negative and reverses sign. An ideal class
ical gas behavior is attained only at temperatures of the order of the
interaction energy, more than an order of magnitude larger than the F
ermi energy.