What could be the reason the oil gauge goes from cold to hot and stays there even though there is not oil issue

Hello, if you truly have an oil temperature gauge -- which does surprise me -- then, it's a cosmetic situation. Oil temperature is not likely to be an issue for your vehicle. If the engine itself is overheating, your coolant temperature gauge will go high and will be a cause for attention. If you want, just ensure you have proper oil pressure. Have an oil pressure gauge hooked up to the oil pressure sending unit port and measure to ensure your engine has sufficient oil pressure on a cold start up, and at normal operating temperature.