We report on the resistance behavior of bare-chip Cernox thermometers under pressures up to 2 GPa, generated in a piston-cylinder pressure cell. Our results clearly show that Cernox thermometers, frequently used in low-temperature experiments due to their high sensitivity, remain highly sensitive even under applied pressure. We show that these thermometers are therefore ideally suited for measurements of heat capacity under pressure utilizing an ac oscillation technique up to at least 150 K. Our Cernox-based system is very accurate in determining changes in the specific heat as a function of pressure as demonstrated by measurements of the heat capacity on three different test cases: (i) the superconducting transition in elemental Pb (Tc = 7.2 K), (ii) the antiferromagnetic transition in the rare-earth compound GdNiGe3 (TN = 26 K), and (iii) the structural/magnetic transition in the iron-pnictide BaFe2As2 (Ts,N = 130 K). The chosen examples demonstrate the versatility of our technique for measuring the specific heat under the pressure of various condensed-matter systems with very different transition temperatures as well as amounts of removed entropy.