It's a great question, but it's not a concern.
In other words, say the regulator is set to 10 PSI. That is regulating the equilibrium of pressure between the room temp gas in the tube and in the tank. But, as the gas moves further down the line, into the fridge, and into the keg, it gets colder. So, is it safe to assume that the keg is 10 PSI if the regulator is at a different temp?
In order for there to be a pressure drop along the gas line, there has to be an active flow of CO2. When you're actively and continuously flowing a significant amount of CO2 to the keg (unless there's a leak this is pretty much only while you have the faucet open), the pressure inside the keg might be slightly less than the regulator gauge reads, but this is due more to the resistance (drag) inside the gas line than temperature. A big change in the temperature of gas in the line might cause the pressure to change a tiny bit, but again only while gas is actively flowing. Chances are it's nothing you'd ever notice, and as soon as you close the faucet it's going to equilibrate almost immediately. If you wanted to do an experiment, you could place a pressure gauge on the cold keg and compare it to the gauge on the warm regulator. But you'd have to make sure the two gauges give identical readings first

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Again, it's not possible for there to be a pressure drop if there's no flow, and it's not possible for there to be flow unless there's a pressure drop to drive it. Normally you open the faucet, the pressure in the keg drops, and CO2 flows through the regulator to increase the pressure back to your setpoint.
An extreme temperature change might cause enough of a pressure drop to create flow, but in a tiny system like this it's not really going to happen. If you connected two huge tanks full of compressed CO2 and threw one of them in a freezer, the pressure would drop in the cold tank, CO2 would flow from the warm tank to the cold tank to equalize the pressure, and then there would be no flow. Everything would be at the same pressure.
There's also resistance in the regulator's diaphragm, which is why when you're flowing a good bit of gas the pressure at the regulator will drop, but then it comes back up as the flow stops. For more critical applications you buy regulators with different Cv ratings depending on your desired flow. But in a kegerator, there's a trickle of CO2 going in as you carbonate but otherwise the only significant flow of CO2 is when you purge the keg or pour a beer. And this isn't the same scenario you're worried about.
I've mentioned this before and it bothered a few people, but Ohm's Law (V = I*R) is universally taught to engineers as an excellent analogy for, among other things, fluid flow. Pressure and voltage are very analogous, as are flow rate and current. The CO2 line can be thought of like an electrical circuit: in order for there to be a voltage drop through a resistor, there needs to be current flowing. And in order for there to be a pressure drop through the tube, there needs to be CO2 flowing. The electrical analogy is often a lot easier to visualize for someone that's relatively new to fluid mechanics.
Yuri makes a good point about the tank side pressure changing with temperature, but it's important to note since highly compressed CO2 is generally stored as a liquid, not a gas, the tank pressure is a function of the CO2 vapor/liquid equilibrium behavior, not gas laws.