preposition omission before "this time of day"

I love going out at this time of the day.

I love going out this time of the day.

Are both the sentences grammatically correct? Or does the omitting the preposition render the sentence ungrammatical?

Answer

Both are correct. In English certain nouns and standard noun phrases that refer to a specific time can function as adverbs without needing to be introduced by a preposition. They indicate when the action or state named by the verb occurred. With some of these phrases, a preposition is not even allowed. These constructions are all fine even in formal English. Some more examples:

I'll call you Tuesday.

I'll call you on Tuesday.

We vacationed in Hawaii last year.

Let's have lunch next week.

This time, I'm going to remember to set the parking brake.

Next time you go to Poppy's Restaurant, order the linguine.

I missed breakfast this morning.

I'll see you tomorrow evening.

Prague is beautiful this time of year.

Prague is beautiful at this time of year.

Next year, we should meet in Jerusalem.

There is no rule for this, though. These are ungrammatical:

We vacationed in Hawaii 2018.

Call me 9:00.

You have to say:

We vacationed in Hawaii in 2018.

Call me at 9:00.