The behavior HTS materials and devices at overload currents is determined in a very high level by cooling, particularly, by the transient heat transfer from HTS device to liquid nitrogen. Recently we observed sharp peaks in the voltage during rectangular current pulse. In the present paper we explain these peaks by analysis of transient regimes in liquid nitrogen. The sharp peak in the voltage is occurring after temperature exceeds certain level before nucleate boiling starts. Basing on measurements of the voltage and temperature of HTS tapes during current overload and numerical analysis of the process we estimated overheat temperatures of nitrogen in transient regimes. Influence of covers of HTS tapes on overheat temperature of nitrogen is discussed. Experimental data and theoretical calculations' are compared.