To put this in simple terms, you have nothing to worry about.
As said above anything above 12.7V once the charger stops charging is surface voltage created by the charger. It will slowly dissipate to its rest voltage of 12.7V if the battery is fully charged.
Look at the chart EMD_DRIVER posted. This is state of charge voltage chart at resting voltage. Rest voltage is the battery voltage while not being charged or being used, drawing no current from it. The battery may have to sit unplugged for a few hours to measure its true resting voltage.
While you are using the battery at night to power your TV, laptop or whatever, the battery will measure below its resting voltage. Once you remove all of the loads the voltage will rise back up. Put a load back on it and the voltage will start going down.
As the above chart shows you can drain the battery down to 12.06V (50%) Rest Voltage and be fine. Measuring 12.5V while running your TV, laptop, lights etc is certainly fine. As stated above, you can go down to 12.2V and still be in good shape. Measuring 12.6V in the morning says your battery is still 90% charged or better. You have barely used its capability at all. Your doing good.