Though if one reads some hadith one may find text that demands death penalty for apostate, but fact is that, no hadith can supersede Quran and Sunnah [actions of Prophet (SAW)], as when hadith contradicts Quran and/or sunnah, than there is no doubt such hadith is fabricated. According to Quran, and also one historical event regarding action of Prophet (SAW), there is no death penalty for apostate in Islam.
First, let discuss the topic from Quran: According to Quran, it is possible that a person accepts Islam (becomes Muslim), then reject Islam (become apostate --- or becomes Kafir again), then become Muslim again (re-enter fold of Islam), then reject Islam again ... and so on can do this yo-yo multiple times (according to Quran, by doing such, they keep increasing their unbelief). Now, if there was death penalty than it was not possible that such could happen and since Quran mentions that this could happen, that means ... death of apostate is not Islamic ... and thus those who claim (or say) that Islam demands death for apostate are rejecter of Allah’s words in Quran. Here is verse from Quran:
Surah Al-Nisa (4), ayah 137: YUSUFALI: Those who believe, then reject faith, then believe (again) and (again) reject faith, and go on increasing in unbelief,- Allah will not forgive them nor guide them nor guide them on the way.
Further, same is mentioned in Quran again ... regarding people who reject faith (Islam) after they accepted it (3:86)... Allah do not guide them ... EXCEPT ... for those that repeat after that and make amends, for verily Allah is oft forgiving and most merciful (3:94) ... shows that Allah accepts their re-entering Islam (that was not possible if their punishment was death in the first place). Further, Quran also talks about those who reject faith (Islam) after they accepted it, and then go on adding to their defiance of faith ... and so on ... So, from these ayah also it is proved that these people are not killed and thus it means Islam does not say that apostate should be killed.
Surah Imran (3), Ayah 86-91:
YUSUFALI: How shall Allah Guide those who reject Faith after they accepted it and bore witness that the Messenger was true and that Clear Signs had come unto them? but Allah guides not a people unjust.
YUSUFALI: Of such the reward is that on them (rests) the curse of Allah, of His angels, and of all mankind;-
YUSUFALI: In that will they dwell; nor will their penalty be lightened, nor respite be (their lot);-
YUSUFALI: Except for those that repent (Even) after that, and make amends; for verily Allah is Oft-Forgiving, Most Merciful.
YUSUFALI: But those who reject Faith after they accepted it, and then go on adding to their defiance of Faith,- never will their repentance be accepted; for they are those who have (of set purpose) gone astray.
YUSUFALI: As to those who reject Faith, and die rejecting,- never would be accepted from any such as much gold as the earth contains, though they should offer it for ransom. For such is (in store) a penalty grievous, and they will find no helpers.
Sunnah of Prophet (SAW): There is not a single confirmed historical event where Prophet (SAW) gave death penalty to anyone who accepted Islam and then became apostate (openly rejected Islam and entered religion of Kafirs again). Actually, such situation did arise in the lifetime of Prophet (SAW) when Abdullah bin Sa’d bin abi Sarh became apostate. His story is that, he not only accepted Islam but became scriber of Holy Quran. Later he went back to his Quraish religion (became apostate) but worse is that he used to joke about Prophet (SAW) and Quran (with lies in his claims). He use to claim that Prophet (SAW) composed Quran himself and it is not revealed.
Before surrender of Mecca, Prophet (SAW) actually ordered about few people who should be killed if they were found, and he was one of them. Anyhow, later when Mecca surrendered to Muslims, Usman (RA) pleaded for his forgiveness and Prophet (SAW) forgave him ... though obviously, if it was order of Allah to kill apostate then Prophet (SAW) would not have forgiven him on plea of anyone, including Usman (RA).
Hence, according to Quran and Sunnah [action of Prophet (SAW)], there is no death penalty in Islam for apostate. This is also obvious as Islam teaches that there is no compulsion in religion, and thus if death penalty for apostate was part of Islam, it would have been compulsion in religion. In Islam, Allah even told Prophet (SAW) that Allah has revealed Quran to Prophet (SAW), but as far as believing on Islam and receiving guidance from Quran is concerned, it is up to individual souls for their own benefits or harms ... and in this matter (religion), not even Prophet (SAW) has any right to dispose their affairs ... whether they believe on Islam, reject Islam, believe and later reject Islam (become apostate), or do whatever they like, as in the end it is they who will die and would get reward or punishment for their deeds in this life.
Surah Az-Zumar (39), ayah 41:YUSUFALI: Verily We have revealed the Book to thee in Truth, for (instructing) mankind. He, then, that receives guidance benefits his own soul: but he that strays injures his own soul. Nor art thou set over them to dispose of their affairs.