The Time Trap Gambit - Agent of T.E.R.R.A., #4

E.M.P.I.R.E., the evil force of time-criminals, never interfered with history unless they had a good reason, and this time their reason was the most devious of all. EMPIRE was altering Earth's time-line, making Rome lose the Second Punic War, because this tampering was certain to lure T.E.R.R.A.'s top agent, Hannibal Fortune. And EMPIRE wanted Hannibal Fortune, wanted him dead. 

Don't miss this classic time travel adventure first published in 1970. 

Here is what Analog Science Fact & Fiction said when it first came out:
"This is the fourth - and by far the best - of the Agent of T.E.R.R.A. time-agent series of T.E.R.R.A. (Temporal Entropy Restructure and Repair Agency) is dedicated to keeping the time threads untangled when others try to remake the past and future to their own
advantage."

This time something is going wrong in the war between Carthage and Rome. The victors are becoming the vanquished - so Hannibal Fortune and his symbiotic buddy Webley are sent hack to 203 B.C. to find out what is happening and see that it is fixed before the history of Planet Earth - Galactic Federation No. 38 - runs wild. He does, he finds the baddies on the scene trying to make Rome lose, and he is suckered into violating some of the basics of time travel, in particular the Rule of Doubletime - no man can exist twice at the same time.