What the fear is usually about
Very few people are frightened of an empty flat. What frightens is the conclusion the empty flat seems to support: that this is permanent, that it says something about your worth, that everyone else managed something you did not.
Separating those is worth doing explicitly, because the fact and the conclusion need different responses. The fact — you live alone now — is a logistics problem. The conclusion is a thought, and thoughts formed at 11pm in month one are poor evidence about anything.
Alone is not lonely
Loneliness is the gap between the connection you have and the connection you want. It is entirely possible to be lonely inside a relationship, and many people reading this were. It is also possible to live alone and not be lonely at all.
The practical consequence: the fix is not 'find someone'. It is closing the gap, and a weekly standing call with one friend often closes more of it than a new relationship would in the same period.
The hours that are actually hard
For most people it is not the whole day. It is the hour after work, Sunday afternoon, and the time between getting into bed and falling asleep. Naming them turns a vague dread of being alone into three specific slots.
Three slots can be planned. Not filled frantically — planned. A standing arrangement on Sunday, something to listen to at bedtime, a route home that does not go straight to an empty room. Modest, repeatable, and considerably more effective than resolving to feel differently.
The cost of avoiding it
The fastest way out of the feeling is someone new, and it works for a few weeks. The difficulty is that the fear is unexamined and comes with you: it reappears as needing constant reassurance, or as staying somewhere unsuitable because leaving means being alone again.
Sitting with it for a few months is unpleasant and buys something real — the knowledge that you can. That knowledge is what later makes it possible to leave a relationship that is not working, which is the same capacity in a different situation.