Is there a specific age at which animals may be slaughtered?

There is no
stipulated age for slaughter of any of the an‘aam animals (camels, cattle,
sheep and goats), if the purpose is only for food. If someone were to
slaughter a lamb that was only one day old or less, it would be
permissible for him to eat that meat, because there is no evidence in
Islam to indicate that this is not allowed, and because the basic
principle concerning what Allah has created for us is that it is
permissible. Anyone who says that something is haraam has to provide
evidence. Allah, may He be exalted, says (interpretation of the meaning):
“He it is Who created for you all that is on earth” [al-Baqarah

2:29]. 

There are
saheeh hadeeths which indicate that the Prophet (blessings and peace of
Allah be upon him) ate lamb. 

Al-Bukhaari
(4101) and Muslim (2039) narrated that Jaabir (may Allah be pleased with
him) said: I said: O Messenger of Allah, come with me to my house. Then I
said to my wife: I have seen something (i.e. hunger) in face of the Prophet
(sa) that caused me concern; do you have anything? She said: I have some
barley and a kid (young goat). So she slaughtered the kid and made dough
with the barley, and we put the meat in the pot. Then I went to the Prophet
(blessings and peace of Allah be upon him) … And he narrated the hadeeth, in
which the Prophet (blessings and peace of Allah be upon him) and his
Companions ate from that kid (young goat). 

According to
the hadeeth of Abu Hurayrah, when the Prophet (blessings and peace of Allah
be upon him) visited Abu’l-Haytham, along with Abu Bakr, and ‘Umar,
Abu’l-Haytham went to make some food for them. The Prophet (blessings and
peace of Allah be upon him) said: “Do not slaughter any animal that is
producing milk. He said: So he slaughtered a kid (young goat) or lamb for
them and brought it to them, and they ate.

Narrated by
Muslim (2038); at-Tirmidhi (2369). 

The word
translated as kid here refers to a female baby goat that has not reached the
age of one year. 

End quote from
an-Nihaayah, Baab al-‘Ayn ma‘a an-Noon. 

The scholars,
may Allah have mercy on them, have stated that if the foetus of an an‘aam
animal comes out of its mother’s womb alive and is slaughtered in the
prescribed manner, it is permissible to eat it. 

Ibn Qudaamah
said in al-Mughni (9/321): If it comes out alive and in a stable
condition, and it is possible to slaughter it, but it is not slaughtered
before it dies, then it has not been slaughtered properly. Ahmad said: If it
comes out alive, then it must be slaughtered properly, because it is another
soul. 

Ibn Nujaym
said in al-Bahr ar-Raa’iq (8/198): If it is known that the sheep was
alive at the time of slaughter, it becomes permissible (halaal) by
slaughtering, whether it moved or not.

End quote. See
also al-Bahr ar-Raa’iq (8/195). 

If it is
proven that Allah, may He be glorified and exalted, has made it permissible
to His slaves to eat that without restricting it to a specific age, then any
condition or restriction that is added to that is an innovation in religion
and is a transgression against what Allah has prescribed for His slaves. 

Allah, may He
be exalted, says (interpretation of the meaning): “And say not concerning
that which your tongues put forth falsely: ‘This is lawful and this is
forbidden,’ so as to invent lies against Allah. Verily, those who invent
lies against Allah will never prosper” [an-Nahl 16:116]. 

And the
Prophet (blessings and peace of Allah be upon him) said: “What is the matter
with men who stipulate conditions that are not in the Book of Allah? Any
condition that is not in the Book of Allah is invalid, even if there are a
hundred conditions. What is ordained by Allah and the conditions stipulated
by Him are more deserving of being adhered to.” 

This hadeeth
was narrated by al-Bukhaari (2168) and Muslim (1504). 

But if the
animal is intended for udhiyah (sacrifice) and the like, then it is
essential that it reach a certain age. This has been discussed in detail in
the answer to question no. 41899.

And Allah
knows best.