The possibilities and limitations of the low-temperature modulated calorimetry are investigated experimentally and theoretically. The effects of the sample holder and of the heat leakage into the thermocouple wires on the measured temperature oscillations are discussed. It is shown that the effects can lead to incorrect measurements and impose limitations on the calorimeter sensitivity and on the width of the appropriate frequency range of the sample temperature modulation. A proper choice of materials and external parameters enables the construction of the calorimeter with sensitivity of ca. 10(-8) J/K at 5 K. The presented calorimeter provides the possibility of the heat-capacity measurements up to 10(3) Hz at low temperatures. High sensitivity and wide frequency range are of principal importance for investigations of complex heat capacity.