Features of the time- and temperature-dependent heat capacity of 1,3,5-tris(l-naphthyl)benzene (T alpha NB) have been studied by differential scanning calorimetry and analyzed in terms of a formalism for the stretched-exponential, nonlinear changes of the fictive temperature, T-f, or enthalpy, H, of its glass. T alpha NB shows a remarkably narrow distribution of structural relaxation times with the stretch-parameter beta of 0.8. The formalism does not fit the data for the irreversible relaxation of its glassy state obtained by rapid cooling. This is attributed to incomplete structural freezing at T-f on rapid cooling of the liquid and a decrease in H, as faster modes of motions also gradually freeze-in on cooling below T-f. Thus, not only H decreases nonlinearly with the temperature but also the heat capacity varies on irreversible relaxation. Therefore, the formalism does not reproduce the structural states when a change in T-f alone is considered. No difference in the reduced temperature width was observed when T alpha NB liquid was rapidly cooled and its glass slowly heated, or vice verse. Samples, annealed or slowly cooled, may show an increase in the width, or a decrease, depending upon the material＇s relaxation characteristics. The reduced temperature width of the glass-softening endotherm seems to provide an unreliable correlation with the temperature dependence of structural relaxation.