I remember in Crash Course World History John Green saying that empires were generally more stable than democracies. Is that true? And if yes, is there a good reason for that?
The issue with this question is that what constitutes an Empire is very uncertain. Most historians would be willing to say that an Empire is some sort of a system comprising vast amounts of territory being controlled largely by a single point. This would be independent of governing system. For example, where does the Athenian Empire fall? Most scholars of the period would agree that by the time of the Peloponnesian War the Athenian Empire was a genuine empire, but at the same time it was a democracy. So how do we categorize that, by this system? And even if we limit Empires to states built around a single individual's power and conquest we run into problems. For example, in what way is Napoleon's French Empire any different from a monarchy? And the British Empire very much was a monarchy. If we take something like the Macedonian Empire under Alexander things get even more complicated. True, Alexander was in sole charge, but the Macedonians didn't have a concept of the Divine right of kings or any concept really that a king had to rule, besides the simple fact that the barons wanted him there. Well the barons had enormous power in such a system, controlling satrapies that were essentially their own semi-autonomous kingdoms and commanding vast armed forces loyal to them alone. So, on a day-to-day level, who was really in charge? Should thst be categorized simply as a monarchy or more as am aristocratic oligarchy headed by an individual with powers vested in him by the trust of his subordinates (of course there's the added difficulty that our modern categorizations of governed often do not apply in antiquity)?
If you want a more straightforward answer, I'd say that if we limit ourselves to empires controlled by single individuals and built up on conquest, which I have a hunch is what you're taking about (although I don't really see how in most cases this is distinct from monarchy), no this is absolutely not the case. Alexander's empire began to fall apart during his own lifetime. Charlemagne's empire collapsed the moment he died. Genghis Khan's empire similarly self-destructed to be partitioned up once he was incapable of policing his subordinates. On the other hand we see the Athenian Empire perpetuating itself for nearly a century, despite the political chaos in which Athens could find herself occasionally. And there's also the Venetian Empire, which could be characterized as a republic or an oligarchy depending on how we look at it. The Venetian Empire lasted quite a lot longer. There are a great deal more factors at play when we discuss the stability and success of any state than the realities of its political system, realities which are also not easily explained away simply
Comparing empires to republics/democracies is comparing apples to oranges.
Republics/democracies are forms of government. An "empire" is the effect of the foreign policy of the government. If a government pursues an expansionist foreign policy, then it will be seen as having imperialist characteristics.
A more direct comparison is between autocracies and democracies/republics. Not all empires were autocratic but many were.
Assuming John Greene was taking about internal stability, and not external stability, it sounds like he might have been referencing the fact that over the life of the state, democracies and republics must respond to the changing values of the people and with this changing values, there is often internal instability. Autocracies generally don't care about the values of the people and so these social changes and their accompanying instabilities are suppressed.
Additionally, all new democracies go through a violent and turbulent period in which competing groups, championing different religious, class, ect issues, try to take power in the new power vacuum. The length of this violence and upheaval in the new democracy varies and it generally depends on exit options for the people who were power positions(high ranking military, ect) before the state was democratic and the prevalence and respect for civil liberties in the new state.
John Green's statement is also a bit disingenuous because there is a lot more data on autocracies than on democracies/republics. While autocracies have been found in cultures around the world since the beginning of recorded history, democracies/republics have only become widespread in the last 50 years. In fact as late as 1980, autocracies still made up 70% of world governments. Statistics on Democracy
As a interesting side note, democracies never go to war with each other(there are many theories on why, no consensus yet). So though democracies exhibit some violent tendencies early in their history and sporadically through their lives, as democracy becomes more and more prevalent, the external stability(war) between states should decrease.