We consider the quantum birth of a hot FRW universe from a vacuum-dominated quantum fluctuation with admixture of radiation and strings which corresponds to quantum tunnelling from a discrete energy level with a non-zero temperature. The presence of strings with the equation of state p = -ε/3 mimics a positive curvature term which makes it possible, in the case of a negative deficit angle, the quantum birth of an open and a flat universe. In the pre-de-Sitter domain radiation energy levels are quantized. We calculate the temperature spectrum and estimate the range of the model parameters restricting temperature fluctuations by the observational constraint on the CMB anisotropy. For the GUT scale of initial de Sitter vacuum the lower limit on temperature at the start of classical evolution is close to the values as predicted by reheating theories, while the upper limit is far from the threshold for a monopole rest mass. © 2002 Elsevier Science B.V. All rights reserved.