The T&P can open if it gets old, but it is designed to open if the pressure is excessive or the temp gets too high. You can measure both fairly easily. Run the hot until it is as hot as it gets into a cup or glass with a thermometer in it and see what it is. You can buy a screw on pressure gauge for about $10 - you can put in on a hose bib, washing machine supply, utility sink, drain line from the WH, or anywhere with adapters. Some have a high pressure hold.
If you have an expansion tank, it could be shot and needs replacing. This will allow the pressure to rise as water in the WH expands. The cold water is denser, when it goes into the tank and then gets heated, it expands with no place to go creating quite high pressures. You could have this situation if you have a check valve in your incoming water lines or have a PRV.
Now, if it is the PRV that is leaking, best to replace it.