It comes directly from the Latin Pompa funebris.

Pompa, itself from the Greek πομπή meant a procession, a parade. We use a the related word pompe in French with the phrase en grandes pompe (English inherited “pompous” from the French pompeux)

Funebris means related to funerals.

Three kinds of pompa (parades) in Rome were:

• the pompa triumphalis (to celebrate a militar victory)
• the pompa funebris (for a funeral)
• the pompa circensis (before circus “games”).

Note also that there are other words sharing the same spelling and which are more common. The first one means “pump” (e.g. pompe à essence is “gas pump”) and the second one is a slang word for shoe (e.g. une paire de pompes: a pair of kicks, à coup de pompes: kicking ass). There is also faire des pompes, i.e. to do push-ups and avoir un coup de pompe, to be exhausted (apparently from an early aviation term where pompe was used to mean the effect of air turbulence.)

There are also regional words like one meaning some kinds of cakes in Southern France and a slang word for cheat sheets.