That strongly depends on what time-frame you're looking at. I only found this study which shows energy usage of refrigerators up to 2002. So unless your fridge is >30 years old, you probably won't see that much of an efficiency gain.

Sadly I can't find a source in English for this, but the German Umweltbundesamt published a study on replacing appliances last year. The study more or less states, that, within a 10 year time frame, it's almost never worth it to replace a fridge from an ecological point of view. It might make economic sense, but that's about it. Here's a short blog post of the Umweltbundesamt about it with some infographics that may be a bit more language-agnostic.

Your fridge (without freezer) would have to use 240 kWh/year for it to make ecological sense to get a new one within the next 10 years. For a fridge-freezer combination it would have to be 340 kWh and for a freezer 430 kWh. From what I saw on this beautiful (also German, but it's basically a huge table anyway) page it depends more on the type of fridge you have, not really the age. Side-by-side fridges seem to be especially bad.