Logopenic variant primary progressive aphasia (lvPPA) is a neurodegenerative language disorder primarily characterized by impaired phonological processing. Sentence repetition and comprehension deficits are observed in lvPPA and linked to impaired phonological working memory, but recent evidence also implicates impaired speech perception. Currently, neural encoding of the speech envelope, which forms the scaffolding for perception, is not clearly understood in lvPPA. We leveraged recent analytical advances in electrophysiology to examine speech envelope encoding in lvPPA. We assessed cortical tracking of the speech envelope and in-task comprehension of two spoken narratives in individuals with lvPPA (n = 10) and age-matched (n = 10) controls. Despite markedly reduced narrative comprehension relative to controls, individuals with lvPPA had increased cortical tracking of the speech envelope in theta oscillations, which track low-level features (e.g., syllables), but not delta oscillations, which track speech units that unfold across a longer time scale (e.g., words, phrases, prosody). This neural signature was highly correlated across narratives. Results indicate an increased reliance on acoustic cues during speech encoding. This may reflect inefficient encoding of bottom-up speech cues, likely as a consequence of dysfunctional temporoparietal cortex.