Which months were added during the Roman Empire?

Your assumption is there was always 365.25 days in a year. But historical records do not agree with that. Particularly before 700 BC there may only have been 360 Days in a year.

The two answers from rancho and EvanM say the Julian calendar was not completed or perfected until during the reign of Augustus the first emperor.

Actually the calendar was in its final, complete, and perfect form when introduced by Julius Caesar in 44 BC but the College of Pontiffs who administered it made what we might consider to be a very basic and obvious mistake.

The Julian calendar has a leap year every fourth year, counted exclusively. If a year is a leap year it is not counted. The next year is year 1, the third year is year 2, the fourth year is year 3, and the fifth year is year 4 and the next leap year. Thus 2012 and 2016 were leap years and 2016 minus 2012 equals four.

But the Pontiffs counted the years inclusively. A leap year would be year 1, the next year year 2, the third year year 3, and the fourth year year 4 and the next leap year. Thus, for example, year 730 AUC and year 733 AUC might both be leap years and 733 minus 730 would equal three, not four.

During the reign of Augustus somebody noticed that the Pontiffs were making leap years too often. Augustus had the practice of adding a leap day halted for several years until the calendar days aligned properly and then had the leap years happen at the right intervals.

Historians are not sure about the exact sequence of leap years and non leap years until the error was corrected. But it is usually believed that leap years since AD 4 or AD 8 were correct. Thus any date in the Julian Calendar after AD 8 at the latest should be correct to the day according to the Julian Calendar.

https://en.wikipedia.org/wiki/Julian_calendar#Leap_year_error[1]

Originally the Calendar was 10 months thanks to King Romulus, The first king of Rome. According to dateandtime.com

The calendar started the year in March (Martius) and consisted of 10 months, with 6 months of 30 days and 4 months of 31 days. The winter season was not assigned to any month, so the calendar year only lasted 304 days with 61 days unaccounted for in the winter.

Later two months were added

The addition of January and February meant that some of the months' names no longer agreed with their position in the calendar (September - December). The month Quintilis was renamed July in honor of Julius Caesar in 44 BCE and Sextilis was renamed August in honor of Augustus in 8 BCE.

It wasn't until Julius Caesar that that we got the Julian calendar but it wasn't perfected until Agustus Caeser's reign.

Resource sited here

The original Roman calendar is believed to have been a lunar calendar, which may have been based on one of the Greek lunar calendars. As the time between new moons averages 29.5 days, its months were constructed to be either hollow (29 days) or full (30 days).

KING ROMULUS -

The original Roman calendar was said to have been invented by Romulus, the first king of Rome, around 753 BCE. The calendar started the year in March (Martius) and consisted of 10 months, with 6 months of 30 days and 4 months of 31 days. The winter season was not assigned to any month, so the calendar year only lasted 304 days with 61 days unaccounted for in the winter.

Calendar of King Romulus:

1. Martius - 31 Days
2. Aprilis - 30 Days
3. Maius - 31 Days
4. Iunius - 30 Days
5. Quintilis - 31 Days
6. Sextilis - 30 Days
7. September - 30 Days
8. October - 31 Days
9. November - 30 Days
10. December - 30 Days

Adding January and February -

The 304-day Roman calendar didn’t work for long because it didn’t align with the seasons. King Numa Pompilius reformed the calendar around 700 BCE by adding the months of January (Ianuarius) and February (Februarius) to the original 10 months, which increased the year's length to 354 or 355 days.

The addition of January and February meant that some of the months' names no longer agreed with their position in the calendar (September - December). The month Quintilis was renamed July in honor of Julius Caesar in 44 BCE and Sextilis was renamed August in honor of Augustus in 8 BCE.

Calendar of king Numa

1. Ianuarius - 29 days
2. Februarius - 28 days
3. Martius - 31 days
4. Aprilis - 29 days
5. Maius - 31 days
6. Iunius - 29 days
7. Quintilis - 31 days
8. Sextilis - 29 days
9. September - 29 days
10. October - 31 days
11. November - 29 days
12. December - 29 days

The Intercalary Month -

The Roman calendar was still flawed after adding January and February, as well as the days and months needed to keep the calendar in line with the seasons. Many attempts were made to align the calendar with the seasons but all failed. An extra month was added to the calendar in some years to make up for the lack of days in a year.

The insertion of the intercalary month was made by the pontifex maximus, the high priest of the College of Pontiffs in ancient Rome. However, this system was flawed because the Roman calendar year defined the term of office of elected officials, thus a pontifex maximus could control the length of the year depending on their political agenda.

When Julius Caesar became pontifex maximus, he reformed the calendar by getting rid of the intercalary months. The Julian calendar was created, then completed during his successor Augustus' reign.

Common Year -

1. Ianuarius - 29 days
2. Februarius - 28 days
3. Martius - 31 days
4. Aprilis - 29 days
5. Maius - 31 days
6. Iunius - 29 days
7. Quintilis - 31 days
8. Sextilis - 29 days
9. September - 29 days
10. October - 31 days
11. November - 29 days
12. December - 29 days

Leap Year -

1. Ianuarius - 29 days
2. Februarius - 23/24 days
3. Intercalaris - 27/28 days
4. Martius - 31 days
5. Aprilis - 29 days
6. Maius - 31 days
7. Iunius - 29 days
8. Quintilis - 31 days
9. Sextilis - 29 days
10. September - 29 days
11. October - 31 days
12. November - 29 days
13. December - 29 days