Does your reputation deteriorate over time?
Solution 1:
Yes it does, with the Major Factions.
According to the Wiki
Reputation decay only applies to the Major factions, Alliance, Federation & Empire.
When your status with a major faction is above 75% Friendly, it will slowly decay down to 75%, by a percentage point every couple of days. Reputation decay happens whether you're logged into the game or not. To stay Allied you have to keep doing good deeds that gain you reputation for the major faction.