We used pressure perturbation calorimetry to investigate the relaxation time scale after a jump into the melting transition regime of artificial lipid membranes. This time is equivalent to the characteristic rate of domain growth. The studies were performed on single-component large unilamellar and multilamellar vesicle systems with and without the addition of small molecules such as general anesthetics, neurotransmitters, and antibiotics. These drugs interact with membranes and affect melting points and profiles. In all systems, we found that heat capacity and relaxation times are linearly related to each other in a simple manner, and we outline the theoretical origin of this finding. Thus, the influence of a drug on the time scale of domain formation processes can be understood on the basis of their influence on the heat capacity profile. This allows estimations of the characteristic relaxation time scales in biological membranes.