  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}$.
