An air-conditioned room with automatic regulation of its microclimate systems using complex control algorithms is one of the most complex objects for calculating non-stationary thermal regime, so this mode is still insufficiently studied. At the same time, such facilities are typical for providing internal weather parameters in civil buildings. In this paper, we consider a simplified mathematical formulation and solution to the problem of changing the temperature of internal air in a room equipped with a back-ground supply ventilation system and automated local cooling systems under variable thermal effects. The main equations connecting the most important components of the heat flow in the room are analyzed, given we neglect the heat accumulation of the array of fences in the first approximation. The dependence on time for the deviation of the room air temperature from the setpoint and the expression for the moment of time at which the maximum temperature deviation is observed, when the heat flow changes abruptly and the equipment of local cooling systems is regulated according to the integral law are presented. Calculations were made to confirm the obtained analytical solution using a finite-difference approximation of the differential equations of heat balance and heat transfer, as well as by comparing the solution obtained by the author earlier, taking into account the spread of the temperature wave in fences, on the example of one of the currently existing residential buildings in the climatic conditions of Moscow. It is noted that the estimated value of the largest temperature deviation from the setpoint (dynamic control error) and the time for this deviation in the considered problem statement do not depend on the transmission coefficient of the regulator. The obtained relations are proposed to be used for an approximate assessment of the non-stationary thermal regime of an air-conditioned room served by local cooling systems controlled by the integral law, as well as for determining the required parameters of the regulator, including using multivariate calculations with changes in the parameters of the problem.