Yes! Here are the things I like to suggest people do when mourning their pets:

1. Post about your pet's death on social media. Some people are hesitant to do this because the feel embarrassed by the grief they feel over their pet or don't want to appear needy, but those I have talked to who have done this have said it has really helped. After posting, friends who are fellow pet people will notice and can reach out and send words of comfort and it's nice to know there are a lot of people you know who have been through this too, who understand. It makes you feel less alone.
2. Figure out some tangible way to hold onto your pet. There are a lot of talented artists (you can find many on Etsy) who will paint a pet portrait, who will make a pet memorial bead mixing some of your pet's ashes into glass, who will make a stuffed animal "clone" of your pet.... these things may seem silly, but having a physical thing to touch and hold when you feel sad helps a lot of people. I hung up my dogs' collars on a nail by the door, which makes me feel like their spirits are still in my apartment.
3. Make a scrapbook of photos of your pet. Print out photos and frame them and keep them around your home. Your pet was part of your life. Don't feel you have to just forget about them now that they're no longer alive. I still have lots of framed photos of my childhood dogs, Gus and Gwen, around my home even though they've been dead for years!
4. Don't necessarily rush into getting another pet. Some people say it helps them immediately to have a new friend in the house, but also a few people I've interviewed have said they wish they had waited a bit before getting another pet because they felt they were constantly comparing the new animal to the old. A nice thing to do can be to volunteer at an animal shelter or offer to dog/cat sit or dog walk for people. Then you can get a bit of an animal fix while still giving yourself space to mourn your pet.
5. Take as much time as you need. It is okay to feel sad and to feel sad for a really long time. I still miss Gwen, and she died in 2013. It's okay to feel sad. Don't rush yourself into feeling better.
6. Reach out for professional help if you are feeling really lost. There are actually a lot of therapists out there now who specialize in pet loss support. (I interviewed a wonderful therapist named Jennifer Breslow who is based in NYC and does art therapy and pet loss support groups.) There are also other pet loss support groups out there which can be a really nice way to find other people who get how you are feeling, and even pet loss hotlines you can call for comfort.