According to Politico, it's because Haftar controls the oilfields among other things. Is it a legitimate fear that Haftar would cut France off oil otherwise? Or is it a strategy for quicker peace?
Each time you see some big powers getting involved in some desolate places, you have to be sure that they have some business there. E.g. siphon off natural resources, or selling arms. Or to protect their allies/stooges which give them access to those resources or sell their arms.
Libya has the largest oil reserve in Africa. In my opinion, France actually doesn't support Haftar. France is pissed with Turkish presence in Northers Libya, and France is finding no way to remove Turkey who sits on GNA like the old man of the seas. Turkish presence disturbs Frech oil ambition in Libya.
This link says that France has a long history of cooperation with Libya including defense product sales and oil. France also has business interests in the post-Gaddafi period.
In May 2016, French engineering firm Technip announced plans to upgrade a major oil platform in a deal worth $500 million. The platform, located north of Tripoli at the Bahr Essalam oil field is capable of producing 12.6 million barrels a day.
So, France initially tried to resolve the civil war by arranging talks with different factions.
In the meantime, taking advantage of the situation in the Mediterranean and Libya, Turkey chipped in by signing a clever pact with the UN-recognized regime(GNA). Then Turkey supplied a lot of arms and GNA pushed Haftar away.
Turkish intervention in Libya is against French interest in that France lost oil exploration possibilities and arms sales in north-west Libya.
Note: Although a Turkish mouthpiece, this video gives a good summary.
Edit (9-July-2020): looks like my theory is proven correct.
In addition to the other answers, it should be considered that stopping the unauthorized flow of migrants to the EU from Libya is a priority for the EU. Given the chaos in Libya, Haftar found some takers for his pitch to be the next strongman who will bring order at the barrel of a gun. In such case, the Libyan migration route would be effectively shut down.
According to the article posted by OP:
A senior French figure familiar with government policy said support for Haftar is partly driven by the imperative of stanching the supply of arms and funds to jihadist groups threatening fragile governments in Niger, Chad and Mali, which are backed by France’s Operation Barkhane.
But the French official said Paris’ love for the Libyan strongman is far more about strategic alliances across the wider Middle East than commercial considerations. Paris is aligned with the Emirati, Saudi and Egyptian rulers, to whom it has sold billions in weapons, against a looser alliance of Qatar, Turkey and the transnational Muslim Brotherhood movement that briefly governed Egypt before being ousted in a military coup in 2013.
French policymakers connect this regional struggle with their fight against Islamist insurgency in the Sahara-Sahel belt and terrorism at home, their No. 1 national security priority, especially since the November 2015 Paris attacks that killed 130 people.
After the instability unleashed by the Arab Spring uprisings, the dominant view in government circles in Paris is that strongman solutions are the only way to keep a lid on Islamist militancy and mass migration, and tant pis (tough luck) for human rights and democracy.