Is it true that Nabi Sallalahu Aalaihiwasallam used to wear topi the whole day except during sleep ?

There are many Ahaadith in support of the Topi (hat). Just as other Sunnats are to be followed at all times, similarly the topi should be worn at all times.

Special care should, however, be given to covering the head when performing Salaat and when answering the call of nature. Hereunder are a few Ahaadith regarding the Topi of Nabi (Sallallaahu Alayhi Wasallam):
1. Aaisha (Radhiallaahu Anha) mentions that when on journey, Nabi (Sallallaahu Alayhi Wasallam) would wear a Topi with ear-flaps. While at home, he would wear a Syrian Topi. (Akhlaaqun-Nani pg.104)
2. Ibn Umar (Radhiallaahu Anhu) says, Nabi (Sallallaahu Alayhi Wasallam) would wear a white Topi. (Tabrani; Majma-uz-zawaaid vol.5 pg.21) 3. Habib ibn Salih (Radhiallaahu Anhu) narrates that when Nabi (Sallallaahu Alayhi Wasallam) would enter the toilet, he wore shoes and covered his head. (Bayhaqi vol.1 pg.96)
4. Waail ibn Hujr (Radhiallaahu Anhu) mentions that he performed Salaat with Nabi (Sallallaahu Alayhi Wasallam) and the Sahaaba (Radhiallaahu Anhum) who were wearing barnases (a type of topi or garment with a hood). (ibn Khuzayma pg.233)

and Allah Ta'ala Knows Best

Mufti Ebrahim Desai