The name Augustus itself is more of a title than a name- it means something like ‘sacred’ or ‘majestic’, and was granted to him by the Roman senate after his victory over Marc Antony and Cleopatra. Here's how the Roman biographer and historian Suetonius described his death:
Supremo die identidem exquirens an iam de se tumultus foris esset, petito speculo, capillum sibi comi ac malas labantes corrigi praecepit, et admissos amicos percontatus [est] ecquid iis videretur mimum vitae commode transegisse... Omnibus deinde dimissis, dum advenientes ab urbe de Drusi filia aegra interrogat, repente in osculis Liviae et in hac voce defecit: "Livia, nostri coniugii memor vive, ac vale!" sortitus exitum facilem et qualem semper optaverat...
On his last day, he would ask now and then if there was any disturbance in the forum on his account, and calling for a mirror, he ordered his hair to be combed, and his hollow cheeks to be adjusted and he enquired of his friends, who were there, if he seemed to them to have performed life's play well enough... Then, having dismissed them all, while he was questioning some who had just arrived from the city, about Drusus's sick daughter, he suddenly died, amidst the kisses of Livia, and with this cry: "Livia! Live with the memory of our marriage; and now, farewell!" having been granted an easy death, and of such a kind as he had always wished for.
[Suetonius, Divus Augustus 99]