Latch up. Several things can make an LM34/35 latch up, one of them being to pull the output down, as the datasheet suggests to allow reading negative temperatures; sequencing matters. Latchup is less likely if Vcc is below about 6 volts, but it can still happen, especially if the output is pulled low statically or by a spike, like from an ADC mux.
These gadgets also like to oscillate with the improper load. And the internal pulldown is very weak, so mux spikes can pull it up and then it'll settle back down slowly.
Bob Pease told me he'd fix it, about 15 years ago. I'm still waiting.
Yup. Lately I've been using the SPI parts, which work great once you write the (expletive deleted) code, or a surface-mount platinum RTD.
John