Cats sleep two-thirds of their lives with 〜そうだ

Grammar Explanation: Hearsay with 〜そうだ

It’s the height of summer in Neo-Venezia. Akari wonders where President Aria is, and they find him sprawled out, sleeping in the one cool spot.

アリシア:: 「猫(ねこ)は一生(いっしょう)の内(うち)　3分(ぶん)の2の時間(じかん)は寝(ね)ているそうよ」; “They say cats spend two-thirds of their lives asleep.”

Key Points

1寝(ね)ているそうよ reports something heard

そうだ after a complete plain-form clause means “I hear that…” or “they say that…”. Alicia didn’t work this out herself. She’s passing along a bit of trivia about cats.
In soft casual speech, the だ drops before よ, giving そうよ. よ presents it as news for Akari.
Compare 寝(ね)そう, with そう on the verb stem. That would mean “looks like he’s about to fall asleep”, a guess from appearance.

2一生(いっしょう)の内(うち) sets the whole

一生(いっしょう) is a whole lifetime. 内(うち) here means “within” or “out of”, so 一生(いっしょう)の内(うち) is the whole span the fraction is taken from.

3Fractions with 〜分(ぶん)の〜

Japanese says the denominator first. 3分(ぶん)の2 is “two out of three parts”, or two-thirds.
3分(ぶん)の2の時間(じかん) is “two-thirds of the time”. The は after it sets that portion of time apart: for that much of their lives, cats are asleep.

4寝(ね)ている as a general trait

寝(ね)る is “to sleep” or “go to bed”. 寝(ね)ている describes being asleep, here as something cats habitually do over their lives, not one nap in progress.

一生 (Lifetime): 一生(いっしょう) is the whole span of one’s life, from birth to death.
内 (Within): 内(うち) is the inside of something. After a period or group, it means “within” or “out of”, as in 一生(いっしょう)の内(うち), “out of a whole life”. It can also mean one’s home or family, usually written うち.
分 (Share): 分(ぶん) is a share or portion, as in 俺の分, “my share”. It also means one’s place or status.
時間 (Time): 時間(じかん) is time, or an amount of time. It also means an hour, or a set time for something.
寝る (To sleep): 寝(ね)る is to lie down and sleep, or to go to bed.
そう (So): そう means “like that” or “that way”. Said alone, it agrees: “right”. After a verb stem or adjective, it means something looks or seems a certain way.