an arbitrary number of plane parallel layers of magnetodielectrics [22, 23]. As was remarked in the familiar review [24], in most of cases the contribution of magnetic properties of the bodies into the van der Waals interaction is very small. It was mentioned also that in some cases, for example, for polarizable particles with both electric and magnetic polarizabilities [25], the inclusion of magnetic properties may be interesting. This was confirmed in the investigation of the impact of magnetic properties of both atoms and material of the wall on atom-wall interactions including the case of multiple walls [22, 26, 27]. Calculation of the influence of magnetic properties of plate materials on the Casimir interation between two magnetodielectric plates was performed in Ref. [28] using the ap- proximation of frequency-independent ε and µ . Repulsive forces were found in a wide range of parameters, and the importance of this phenomenon for experimental study and for nanomachinery applications was noted. It was shown [29], however, that for real materials µ is nearly equal to unity in the range of frequencies which gives major contribution to the Casimir force. As a result, the magnitude of µ is always far away from the values needed to achieve the Casimir repulsion [29]. In this connection, Ref. [23] reconsidered this problem for the configuration of one metal and one magnetodielectric plate taking into account dependences of ε and µ on the frequency. In so doing the metal (Au) was described by the Drude dielectric permittivity and the permittivity and permeability of a magnetodielectric was described by a simplified model of the Drude-Lorentz type. It was shown that at zero temperature there is a repulsive regime, but only at large separations of about 15 µ m. At nonzero temperature the Casimir force was found to be always attractive. It should be taken into account, however, that at room temperature the theoretical description of the Casimir force by means of the Lifshitz theory combined with the Drude model is experimentally excluded at high confidence level [14, 30, 31]. Some authors [32], however, called for the reanalysis of electrostatic calibrations in previous experiments on the Casimir force basing on their measurements with by a factor of 200 larger sphere radius. This call initiated a discussion in the literature [33, 34]. Further study may be needed here before the situation will become well understood. Because of this, it is worthwhile to analyze the problem by using different approaches to the theory of thermal Casimir force suggested in the literature with allowance made for all existing types of magnetodielectric materials. In this paper we investigate the thermal Casimir force between magnetodielectric plates 3