The gemara says that Yerov'am was justified in giving mussar to Shlomo HaMelech for closing the gates of those going up to Yerushalayim for Yom Tov for the sake of doing "angarya" for the daughter of Pharoh. Rashi explains that Shlomo was trying to force the Jews to either pay taxes or tribute to her on their way in to Yerushalayim. But, Rashi offers a 3rd peshat - Shlomo would lock the doors of the courtyard of the Beis HaMikdash and hold the key. He would generally sleep until the 3rd hour of the day and the Jews would patiently wait for him to open the doors before sacrificing the morning Tamid, and Yerovam accused him of wanting payment toward Bas Pharoh as a condition to hand over the keys.
I once heard a shiur from Rav Nota Greenblatt where he explained a Rambam in Temidin U'musafin (1:2) who writes about the appropriate time for the sacrificing of the Tamid -
ואימתי זמן שחיטתן של בקר? שוחטין אותו קודם שתעלה החמה משיאור פני כל מזרח, ופעם אחת דחקה השעה את הצבור בבית שני והקריבו תמיד של שחר בד' שעות ביום
At first the Rambam writes that the Tamid must be brought when the Eastern side lights up, no later, but then goes on to quote a story in the second beis hamikdash where it was brought later. After much elaboration and build up to explain the concept, Rav Nota explained that by korbanos the kashrus of the korban is not necessarily dependent on the fulfillment of the mitzvah. Regarding the mitzvah, the last moment to bring the korban is משהאיר המזרח, but regarding the kashrus of the korban it is still valid until the 4th hour of the day. Therefore, the Rambam quotes the situation where it was once brought late, indicating that it is still kasher, but the mitzvah has been lost. He supported this argument based on a Rabbeinu Yona in Brachos. Rabbeinu Yona (4th perek 18 in pages of Rif) on the testimoney of Rav Yehuda Ben Bava about the Tamid being brought at the 4th hour, writes -
ואומר במדרש שהמלך שלמה היה ישן עד ג' שעות כמו שדרך בני מלכים לעמוד בג' שעות, ופעם אחת הטעתו בת פרעה וישן עד ד' שעות והקריבו תמיד של שחר בד' שעות, ומזה היתה קבלה ביד ריב"ב והעיד שתמיד של שחר קרב בד' שעות
Rabbeinu Yona explains that the testimony of Rav Yehuda was based on the story of Shlomo where it was once actually brought at the 4th hour. The Hagahos Ma'aseh Ilfas (printed after the Rif in the vilna shas) asks that the midrash continues that Shlomo was harshly rebuked by his mother Bas Sheva, or according to some versions was rebuked by Yerovam Ben Nevat, implying that it was against the din to sacrifice the Tamid so late in the day. This midrash is also quotes by rashi in Mishlei 31 - when shlomo married bas pharoh on the day of the chanukas beis hamikdash, he stayed awake all night listening to music and therefore over slept the next morning until the 4th hour. Since the keys were under his pillow, the Tamid was brought in the 4th hour and Bat Sheva rebuked him for this. How then can we learn from here that it is permissible to sacrifice the Tamid until the 4th hour - this was a mistake of Shlomo's for which he was rebuked? Rav Nota explained that this could have been the Rambam's source. In the 4th hour the korban is still kasher, yet Shlomo was rebuked by his mother because the time that the mitzvah of bringing the korban can be fulfilled had expired.
The difficulty with this approach is that both Rabbeinu Yona and Rashi by us seem to describe that had Shlomo woke up at the 3rd hour, as he routinely did, he would not have received rebuke. Yet, according to this approach in the Rambam, the z'man mitzvah had already passed when the Eastern side lit up (way before the third hour). Furthermore, if the mitzvah of Tamid cannot be fulfilled in the 4th hour, why does Rav Yehuda hold that one can daven in the 4th hour? Shouldn't the mitzvah of tefilah be bound to the mitzvah of the Tamid rather than just the kashrus of the Tamid?