The suppression of core temperature fluctuations due to edge cooling in the J-TEXT tokamak is presented. The supersonic molecular beam is injected into the low density plasmas and the core temperature rise appears. The staggered time correlation technique is utilized to infer the temperature fluctuations from an electron cyclotron emission radiometer (ECE). The existing correlation ECE (CECE) also enables the observation of the temperature fluctuations simultaneously. A significant flattening of the density gradients appears almost in the whole plasmas after the cold pulse injection. The behaviors of the temperature fluctuations from both ECE and CECE show a good agreement, except for the fluctuation level, which exhibits a relative variation of approximately 10-50%. The relative intensity of temperature fluctuations decreases within the surface of r/a~0.8 due to the edge cooling. The cross-power analysis and the agreement in measurements suggest that temperature fluctuations from turbulence inferred from a single ECE channel may be possible. Linear simulations suggest the drop in temperature fluctuation intensity may relate to the trapped electron mode turbulence stabilization. This paper also demonstrates the evolution of both gradients and fluctuations before and after supersonic molecular beam injection with density scans.