We present the results of Suzaku and Chandra observations of the galaxy cluster RXC J1053.7+5453 ($z=0.0704$), which contains a radio relic. The radio relic is located at the distance of $sim 540$ kpc from the X-ray peak toward the west. We measured the temperature of this cluster for the first time. The resultant temperature in the center is $ sim 1.3$ keV, which is lower than the value expected from the X-ray luminosity - temperature and the velocity dispersion - temperature relation. Though we did not find a significant temperature jump at the outer edge of the relic, our results suggest that the temperature decreases outward across the relic. Assuming the existence of the shock at the relic, its Mach number becomes $M simeq 1.4 $. A possible spatial variation of Mach number along the relic is suggested. Additionally, a sharp surface brightness edge is found at the distance of $sim 160$ kpc from the X-ray peak toward the west in the Chandra image. We performed X-ray spectral and surface brightness analyses around the edge with Suzaku and Chandra data, respectively. The obtained surface brightness and temperature profiles suggest that this edge is not a shock but likely a cold front. Alternatively, it cannot be ruled out that thermal pressure is really discontinuous across the edge. In this case, if the pressure across the surface brightness edge is in equilibrium, other forms of pressure sources, such as cosmic-rays, are necessary. We searched for the non-thermal inverse Compton component in the relic region. Assuming the photon index $ Gamma = 2.0$, the resultant upper limit of the flux is $1.9 times 10^{-14} {rm erg s^{-1} cm^{-2}}$ for $4.50 times 10^{-3} {rm deg^{2}}$ area in the 0.3-10 keV band, which implies that the lower limit of magnetic field strength becomes $ 0.7 {rm mu G}$.
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