The Cumbrians have lost all three league games, conceding 13 goals, and Abbott admitted after Saturday's 4-0 home defeat to Coventry that he and assistant Graham Kavanagh had reached rock-bottom.
"I don't want to be gardening or shopping on a Saturday," Abbott said at his weekly press conference.
"I want to be in a dug out and I want to be working with football people. This is a determined character, who is ready to fight for what I believe in.
"I think it was 52 managers and their staff who got sacked last year. That's over half, so it is a tough business.
"At the rate I'm going, it can't go on too much longer like this - that's obvious to everyone.
"(Managing director) John (Nixon) has been sympathetic because he knows the situation, but he has said that things have to improve.
"He's a decent fella and he's seen that there is a remedy there. There's no room for excuses now that the main bulk of the group is back because we just have to be better.
"I haven't been given an ultimatum because pressure just brings decreased performance.
"They (the board) might have an ultimatum in their own heads, but what's the point in telling me?
"I can't try any harder than I already am, and I know what happens with the industry we're in. I don't need anybody to tell me we aren't playing well. I know that.
"I'm in the firing line and that's the industry we work with."
Carlisle will attempt to stop the rot at unbeaten Colchester this weekend.