The pasuk says in the first perek of Yeshayahu that when the Jews were sinning מנחת שוא קטורת תועבה היא לי - The korbanos mincha that they were bringing were in vain and their ktores was an abomination. The Meshech Chochma [haftora to Parshas Devarim] wondered why specifically these offerings?
He explained that a mincha cannot be brought from partners [אינה קריבה בשותפין] but only from an individual or community [which is one unified unit]. Since the Jews had lost their sense of unity and togetherness, the menachos were in vain. Also, ktores is incense where all of the herbs combine to make a pleasant smell - including the chelbana which was foul smelling. Since the Jews had lost their sense of togetherness, their ktores was an abomination because the chelbana was alone and "stank up the place" [PLEASE forgive the expression]. עיין שם in the Meshech Chochma.
The problem is that this pasuk is talking about the first Beis Hamikdash was was not destroyed because of sinas chinam as the second one was?
The Rinas Yitzchak cites the Kli Yakar at the beginning of Dvarim who says that in Bayis Rishon there was also sinas chinam - among the leaders, while in Bayis Sheni it spread to the people. The Maharal says [Netzach Yisrael 5] that the job of the king is to unite the people. So during Bayis Rishon the leaders were not trying to unite the people but were interested in their own power and position which ultimately filtered down to the people. That is what the pasuk means when it says in the first perek of Yeshayahu שמעו נא קציני סדום - The leaders of the Jews were like leaders of Sodom. So the Meshech Chochma is correct in explaining that during Bayis Rishon there was sinas chinam.
We see the destruction wrought by leaders who are more concered with their own personal power than they are about the good of the nation. בינו נא זאת ולא אוכל להאריך במסגרת זו.
BS"D I thought to explain differently: It is clear that the Jews lacked unity during Bayis Rishon as well [and the Meshech Chochoma is correct]. What were the three sins for which the first Beis Hamikdash was destroyed? Gilui Arayos, Shfichus Domim and Avoda Zara.
Gului Arayos is a symptom of tyvos and the pasuk says לתאוה יבקש נפרד - Those who are immersed in their base desires seek separateness. In addition, the pasuk in Amos teaches that the Jews were involved in wife-swapping [see Amos 6/4 and Kiddushin 71b. It is actually quite unbelievable but the reality is that most of what goes on today was also going on thousands of years ago. They just didn't have the Internet to help them along]. Sleeping with someone else's wife makes them forbidden to each other forever. Comming such a sin can't come from such a great place of Ahavas Yisrael.
Sfichus Domim - Killing each other doesn't sound like they were getting along so well.
Avoda Zara - The entire idea of Avoda Zara is a competition of different gods for power. The essence of Avoda Zara is פירוד - separation.
So during Bayis Rishon people weren't getting along either. The gemara says that the first Bayis was destroyed because of the three cardinal sins because they are halachically much worse than mere hatred but the hatred was clearly there too.