It depends if your state has minimum energy requirements for hunting deer. Some do and some don't allow deer hunting with air guns at all. It also depends on your budget as there is a fairly broad range.
There are two levels or categories of big bore too. You have the relatively low powered mainstream .30 to 50 caliber guns that put out 80fpe to 400fpe. Then you have the extreme big bores which typically shoot cast bullets from .257 up to 72 caliber and up to 1700fpe (using helium)
The mainstream ones are a little easier to live with as many are repeaters, have decent shot counts (for a big bore) and accuracy. At the higher price range, you have guns like the RAW 9mm and the FX Boss 30 cal which are both fairly refined for big bore air guns although they are not particularly powerful (80-100fpe). at the lower price range, you have the Sam Yang single shot big bores like the light hunter, also, the Benjamin bulldog (357) and the Hatsan Carnivore.
To see how these mainstream big bores differ from the extreme big bore, check out a company (conveniently) called "Extrem Big Bores". There are others (like Dennis Quakenbush) but they are as good as any to illustrate the differance. Their guns have only 2 shots per fill but they pack enough power to take big game like bear and buffalo. In some states, they are the only option for hunting larger game with air guns.
Personally, I see no point in big bore air guns but if I did, and I wanted to hunt deer, and my State didn't impose really high minimum power requirements, I would probably look at the American Air Arms 357 Slayer. it's expensive but it produces decent power and it's not single shot.
if you provide a budget and find out your states laws, it will be easier to narrow it down.
