The Animal Domain is problematic...
I know it breaks the suspension of disbelief, but it really isn't much different from the Tumor Familiar discovery, where each day you can create a new familiar from a tumor on your character.
The rules for druid's animal companions are unclear about what happens when a druid loses the companion (by death or other means). Druids, rangers, and hunters do have a spell to call possible companions, Call Animal, but others classes that gain animal companions (cavaliers, samurai, clerics, oracles and others), don't have anything like that. So the player will have to work with the GM for a proper way of obtaining a new animal.
In one of my campaigns, one player made a spell-less ranger, and we ruled that he would have to hunt (through Survival minigames I prepared on the fly), or buy a new animal, and after a week, the animal would already be trained enough to be considered a companion and would learn the free tricks and feats that they normally would get. But this was a class that had a permanent companion, not a temporary one.
Since the cleric/oracle animal companion lack any special ability to call a new animal, you have to assume that they pray for it, and the animal shows up some time after that, whenever it is convenient for the character and the GM to introduce the animal into his story. For the alchemist, its even more difficult, as he doesn't get to pray and ask for anything from his god, as the animal could be summoned from the planes when he drinks the cognatogen, for instance.
In fact, since this is an animal companion and not actually a summoning spell, you could rule that he only has one animal, like most classes, and that this same companion shows up every time he drinks his cognatogen in that same day that he prepared his cognatogen. This is the animal granted to her by her god and it wouldn't make sense to be a new creature every time this ability is used unless there is a good reason for it.
This saves time during play and prevents that the player prepares 20 different companions for every situation he could think of and gain an advantage on the creature's abilities in that specific situation (like a summoning spell). But personally, I would make this a single creature, period. And he would need to do a minor quest to change it or have the previous creature killed, accidentally or during combat, but never on purpose. But this is probably too much house-ruling for you.
On the Pathfinder Society, the developers added the following rule to make things a little more balanced for alchemists that can drink multiple cognatogens in a single day:
Page 11—If a sacramental alchemist uses any domain abilities with a limited number of daily uses, he must select the same domain any time he prepares a sacramental cognatogen for the rest of the day, tracking expended abilities as normal for that domain. For example, if a domain grants a 1/day ability, he cannot use it a second time by rebrewing his cognatogen.
The Ultimate Campaign introduced some guidelines to revive and replace companions, but you will find that they are very limited on mechanics. In the end you will have to figure out something together with the GM. You wouldn't want the alchemist to pray for 24-hours after drinking his cognatogen so he can take benefits of his new companions, right?
Choosing an animal companion requires 24 hours of prayer.
That line is talking about rangers and druids (divine casters), as the only class, at that time, that didn't have a spell to summon their companion was the cavalier. But the next guideline actually sums up this answer:
In terms of game mechanics, there is no difference between any of these options, and you should work with the GM to find a replacement method that is appropriate to the campaign.
Otherwise, how would those characters obtain a divide-granted animal? It's magic.
