I ran Maxent with 10 replicates in cross-validation. I got ten results. Now I am confused which result to take for final model evaluation, the best result on the basis of AICc, TSS, AUC and Omission rate or the average result?
if you make 10 replicates, you should have results for each replication (with name of your species and number, from Species_0 to Species_9), hovewer, you should also have html file with average value for all replications (it is file without any number, like Species.html). Using the average values is better than choosing one replication, especially if you divide your dataset using 'random seed' option. Hovewer, 10 replications are few, it would be good to have at last 100.
Forwarding Łukasz Walas answer, MaxEnt typically offers you the averaged model already. In the case of 10 replicates of f.e. mus_musculus, you would get html files (summarizing the results) for model 0 to 9, as well as a html file called mus_musculus. The models themselves are the ascii files, and the mus_musculus_avg is the one model averaged overall replicates.
Also you should look at the average AUC of the test points (not automatically reported by MaxEnt, but easily computable as you only have to look into the single model html files and extract their AUC).