Overheating 1995 5.0

Hello I recently bought a 1995 mustang gt 5.0 5 speed . The car ran great but recently it's been overheating, I live in phx az so It gets pretty hot. I noticed the temp gauge goes a little bit above the middle at times but it constantly goes up and down little by little but when I floor it the car slowly starts to heat up bad just a little bit before the red line. I'm not sure if I have a bad water pump or radiator. The car is fairly stock. It has a cobra 5.0 manifold, it is tuned and it has 4 inch mufflers that's about it so I find it pretty strange for it to overheat. I need help plz anything helps
 
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Since you're fan's working, water pump, radiator, thermostat, or blown head gasket are your usual suspects for overheating. It may not hurt to check the actual radiator temperature (using a laser thermometer - they're really cheap these days ~$10 at Walmart) just to make sure it's not just your gauge or its sensor acting up.
 
Since you're fan's working, water pump, radiator, thermostat, or blown head gasket are your usual suspects for overheating. It may not hurt to check the actual radiator temperature (using a laser thermometer - they're really cheap these days ~$10 at Walmart) just to make sure it's not just your gauge or its sensor acting up.
I think it might just be the sensor messed up because the car will ride normal for about 5 minutes then it goes up to about midway the M and A in NORMAL then after about a couple minutes it falls back down to normal, all while driving calmly, it happens when it's 98 degrees outside and it happens when it's 75 degresss outside