Slow magnetoacoustic waves are omnipresent in both natural and laboratory plasma systems. The appearance of dispersive properties of the slow magnetoacoustic waves, caused by the wave-induced misbalance between cooling and heating processes is discussed. The wave dispersion could be attributed to the presence of characteristic time scales in the system, connected with the plasma heating or cooling due to the competition of the heating and cooling processes in the vicinity of the thermal equilibrium. The dispersion is manifested by the dependence of the effective adiabatic index of the wave on the wave frequency, making the phase and group speeds frequency-dependent. The wave evolution is shown to be controlled by the dispersion, as well as by the amplification or attenuation associated with the thermal misbalance too. As an example, slow magnetoacoustic waves in the plasma of the solar corona are considered. A broadband wave evolution is found to show typical dispersive behaviour, such as broadening and formation of an oscillatory wake. The dispersive effects are most pronounced in the vicinity of a characteristic frequency determined by the thermal misbalance. The occurrence of this dispersion affects the efficiency of nonlinear cascade, leads to the formation of quasi-periodic wave trains, and modifies the effective values of the adiabatic index and the apparent sound speed.