Firstly, regardless of the legality of your landlord's claim, he's full of shit.

Cat fleas live (as adults) for approximately 1-2 months with an acceptable host. Humans are a sub-prime host because fleas are not able to feed adequately from a human in a natural environment (they will obtain 4-5min worth of feeding from a human host, compared to 10+ minutes of feeding on an animal host). Fleas are also unable to reproduce on humans in a natural environment (laboratory studies have been done which allowed flees to live and breed on humans, but they had to have free access to human blood and be feeding almost 12 hours a day compared to the 4-5min they might manage in a normal environment... they're unable to get enough blood to sustain their life cycle adequately, nor enough blood to be able to lay eggs). This is further compounded by humans not having thick enough hair and fleas being unable to survive through a bath or lengthy shower.

Occam's razor, either the current tenant in that suite has a pet (and thus your landlord has no way of proving whether the original infestation is from your pet or the new animal). The only other possibility is that the apartment doesn't have fleas and your landlord is just trying to squeeze you, which seems unlikely this many months down the road.

Additionally, fleas are really easy to remove from an environment that doesn't have pets. Washing all of the carpets in a suite is usually sufficient, but a simple scattering of diatomaceous earth and a thorough vacuuming is enough to eradicate them as long as there's no pets for them to feed (and breed) off.

I would be tempted to ask your landlord if the current owner of the suite has a pet, but it's likely wiser to simply ignore him unless he files a suit.